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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2411.14397v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S1" title="In Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2" title="In Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Exact solution and spectrum of discrete Schrödinger equation on a finite chain</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3" title="In Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Basic theory of quantum graphs</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4" title="In Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Branched lattices</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.SS1" title="In 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Arbitrary branching topology</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.SS2" title="In 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Star branched lattice</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.SS3" title="In 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Possible experimental realization of the model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S5" title="In Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S6" title="In Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Acknowledgements</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"><span class="ltx_note ltx_role_institutetext" id="id1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_note_type">institutetext: </span> National University of Uzbekistan, Universitet Str. 4, 100174, Tashkent, Uzbekistan, <a class="ltx_ref ltx_url ltx_font_typewriter" href="mashrabresearcher@gmail.com" title="">mashrabresearcher@gmail.com</a> <br class="ltx_break"/> Institute for Mathematics, TU Ilmenau, Weimarer Str. 25, 98693, Ilmenau, Germany, <a class="ltx_ref ltx_url ltx_font_typewriter" href="carsten.trunk@tu-ilmenau.de" title="">carsten.trunk@tu-ilmenau.de</a> <br class="ltx_break"/> Kimyo International University in Tashkent, Usman Nasyr Str. 156, 100121, Tashkent, Uzbekistan, <a class="ltx_ref ltx_url ltx_font_typewriter" href="jambul.yusupov@gmail.com" title="">jambul.yusupov@gmail.com</a> <br class="ltx_break"/> Turin Polytechnic University in Tashkent, Niyazov Str. 17, 100095, Tashkent, Uzbekistan, <a class="ltx_ref ltx_url ltx_font_typewriter" href="dmatrasulov@gmail.com" title="">dmatrasulov@gmail.com</a> <br class="ltx_break"/> Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan </span></span></span> <h1 class="ltx_title ltx_title_document">Discrete Schrödinger equation on graphs: <br class="ltx_break"/>An effective model for branched quantum lattice</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">M. Akramov </span><span class="ltx_author_notes">11</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> C. Trunk </span><span class="ltx_author_notes">22</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> J. Yusupov </span><span class="ltx_author_notes">33</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> D. Matrasulov </span><span class="ltx_author_notes">44551122334455</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schrödinger equation that is used to write the solution for quantum graphs. Formulation of the problem and derivation of secular equation for arbitrary quantum graphs is presented. Application of the approach for the star graph is demonstrated by obtaining eigenfunctions and eigenvalues explicitely. Practical application of the model in conducting polymers and branched molecular chains is discussed.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Modeling of quantum particle motion in low-dimensional branched structures is of importance for many practical applications, especially for the designing and optimization of functional properties of quantum materials. A powerful tool for such purpose is to use quantum graph theory. The advantage of quantum graphs in modeling quantum transport in low-dimensional branched structures and networks comes from the fact that the description can be effectively reduced to a one-dimensional Schrödinger (Dirac) equation on metric graphs. In most of the cases, quantum graphs based approach allows one to obtain an exact solution to the problem for arbitrary graph topologies.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Quantum graphs are determined as the branched quantum wires, which are connected to each other at the nodes (vertices) according to a certain rule called the topology of a graph. Initially, the quantum graphs were introduced for modeling of electron motion in low-dimensional molecular structures in the quantum chemistry of organic molecules. In particular, the Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib3" title="">3</a>]</cite>, where the electron motion in branched aromatic molecules was studied, can be considered as a pioneering attempt. However, the strict formulation of the quantum graph concept, where the latter was defined as branched quantum wire, has been presented a few decades later by Exner and Seba in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib4" title="">4</a>]</cite>. The next step in the systematic study was done by Kostrykin and Schrader, who proposed general vertex boundary conditions ensuring self-adjointness of the Schrödinger operator on graphs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib5" title="">5</a>]</cite>. Later the quantum graph concept has been used in different contexts (see, the Refs.<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib9" title="">9</a>]</cite>) and an experimental realization in microwave networks was done <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib8" title="">8</a>]</cite>. For an overview of different mathematical aspects of the quantum graph theory, we refer to the books <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib11" title="">11</a>]</cite>. Some physically important problems of the quantum graphs theory are studied in the Refs.<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib20" title="">20</a>]</cite>. Different issues of evolution and spectral equations on fractal and discrete systems, including graphs are considered in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib29" title="">29</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">In this paper, we consider a version of a quantum graph with discrete edges, i.e., each branch represents a discrete chain of finite length. Such a system models branched lattices, where particle motion occurs in the quantum regime.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Using the solution of the discrete Schrödinger equation on a finite 1D lattice, we construct a solution of the problem for the discrete quantum graph, which satisfies vertex boundary conditions. Finally, the eigenvalues are determined in terms of a secular equation.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Exact solution and spectrum of discrete Schrödinger equation on a finite chain</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.7">Discrete Schrödinger equation has attracted attention in the context of mathematical physics, exactly solvable models and spectral theory <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib33" title="">33</a>]</cite>. The one-dimensional time-independent Schrödinger equation for a free particle is written as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{d^{2}\Psi(x)}{dx^{2}}+k^{2}\Psi(x)=0," class="ltx_Math" display="block" id="S2.E1.m1.3"><semantics id="S2.E1.m1.3a"><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mfrac id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml"><mrow id="S2.E1.m1.1.1.1" xref="S2.E1.m1.1.1.1.cmml"><msup id="S2.E1.m1.1.1.1.3" xref="S2.E1.m1.1.1.1.3.cmml"><mi id="S2.E1.m1.1.1.1.3.2" xref="S2.E1.m1.1.1.1.3.2.cmml">d</mi><mn id="S2.E1.m1.1.1.1.3.3" xref="S2.E1.m1.1.1.1.3.3.cmml">2</mn></msup><mo id="S2.E1.m1.1.1.1.2" xref="S2.E1.m1.1.1.1.2.cmml">⁢</mo><mi 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xref="S2.E1.m1.1.1.3"><times id="S2.E1.m1.1.1.3.1.cmml" xref="S2.E1.m1.1.1.3.1"></times><ci id="S2.E1.m1.1.1.3.2.cmml" xref="S2.E1.m1.1.1.3.2">𝑑</ci><apply id="S2.E1.m1.1.1.3.3.cmml" xref="S2.E1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.E1.m1.1.1.3.3.1.cmml" xref="S2.E1.m1.1.1.3.3">superscript</csymbol><ci id="S2.E1.m1.1.1.3.3.2.cmml" xref="S2.E1.m1.1.1.3.3.2">𝑥</ci><cn id="S2.E1.m1.1.1.3.3.3.cmml" type="integer" xref="S2.E1.m1.1.1.3.3.3">2</cn></apply></apply></apply><apply id="S2.E1.m1.3.3.1.1.2.2.cmml" xref="S2.E1.m1.3.3.1.1.2.2"><times id="S2.E1.m1.3.3.1.1.2.2.1.cmml" xref="S2.E1.m1.3.3.1.1.2.2.1"></times><apply id="S2.E1.m1.3.3.1.1.2.2.2.cmml" xref="S2.E1.m1.3.3.1.1.2.2.2"><csymbol cd="ambiguous" id="S2.E1.m1.3.3.1.1.2.2.2.1.cmml" xref="S2.E1.m1.3.3.1.1.2.2.2">superscript</csymbol><ci id="S2.E1.m1.3.3.1.1.2.2.2.2.cmml" xref="S2.E1.m1.3.3.1.1.2.2.2.2">𝑘</ci><cn id="S2.E1.m1.3.3.1.1.2.2.2.3.cmml" type="integer" xref="S2.E1.m1.3.3.1.1.2.2.2.3">2</cn></apply><ci id="S2.E1.m1.3.3.1.1.2.2.3.cmml" xref="S2.E1.m1.3.3.1.1.2.2.3">Ψ</ci><ci id="S2.E1.m1.2.2.cmml" xref="S2.E1.m1.2.2">𝑥</ci></apply></apply><cn id="S2.E1.m1.3.3.1.1.3.cmml" type="integer" xref="S2.E1.m1.3.3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.3c">\frac{d^{2}\Psi(x)}{dx^{2}}+k^{2}\Psi(x)=0,</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.3d">divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ψ ( italic_x ) end_ARG start_ARG italic_d italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ψ ( italic_x ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.4">on the infinite line <math alttext="x\in(-\infty,+\infty)" class="ltx_Math" display="inline" id="S2.p1.1.m1.2"><semantics id="S2.p1.1.m1.2a"><mrow id="S2.p1.1.m1.2.2" xref="S2.p1.1.m1.2.2.cmml"><mi id="S2.p1.1.m1.2.2.4" xref="S2.p1.1.m1.2.2.4.cmml">x</mi><mo id="S2.p1.1.m1.2.2.3" xref="S2.p1.1.m1.2.2.3.cmml">∈</mo><mrow id="S2.p1.1.m1.2.2.2.2" xref="S2.p1.1.m1.2.2.2.3.cmml"><mo id="S2.p1.1.m1.2.2.2.2.3" stretchy="false" xref="S2.p1.1.m1.2.2.2.3.cmml">(</mo><mrow id="S2.p1.1.m1.1.1.1.1.1" xref="S2.p1.1.m1.1.1.1.1.1.cmml"><mo id="S2.p1.1.m1.1.1.1.1.1a" xref="S2.p1.1.m1.1.1.1.1.1.cmml">−</mo><mi id="S2.p1.1.m1.1.1.1.1.1.2" mathvariant="normal" xref="S2.p1.1.m1.1.1.1.1.1.2.cmml">∞</mi></mrow><mo id="S2.p1.1.m1.2.2.2.2.4" xref="S2.p1.1.m1.2.2.2.3.cmml">,</mo><mrow id="S2.p1.1.m1.2.2.2.2.2" xref="S2.p1.1.m1.2.2.2.2.2.cmml"><mo id="S2.p1.1.m1.2.2.2.2.2a" xref="S2.p1.1.m1.2.2.2.2.2.cmml">+</mo><mi id="S2.p1.1.m1.2.2.2.2.2.2" mathvariant="normal" xref="S2.p1.1.m1.2.2.2.2.2.2.cmml">∞</mi></mrow><mo 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id="S2.p1.1.m1.2c">x\in(-\infty,+\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.2d">italic_x ∈ ( - ∞ , + ∞ )</annotation></semantics></math>, where <math alttext="\hbar=2m=1" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mrow id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml"><mi id="S2.p1.2.m2.1.1.2" mathvariant="normal" xref="S2.p1.2.m2.1.1.2.cmml">ℏ</mi><mo id="S2.p1.2.m2.1.1.3" xref="S2.p1.2.m2.1.1.3.cmml">=</mo><mrow id="S2.p1.2.m2.1.1.4" xref="S2.p1.2.m2.1.1.4.cmml"><mn id="S2.p1.2.m2.1.1.4.2" xref="S2.p1.2.m2.1.1.4.2.cmml">2</mn><mo id="S2.p1.2.m2.1.1.4.1" xref="S2.p1.2.m2.1.1.4.1.cmml">⁢</mo><mi id="S2.p1.2.m2.1.1.4.3" xref="S2.p1.2.m2.1.1.4.3.cmml">m</mi></mrow><mo id="S2.p1.2.m2.1.1.5" xref="S2.p1.2.m2.1.1.5.cmml">=</mo><mn id="S2.p1.2.m2.1.1.6" xref="S2.p1.2.m2.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><apply id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1"><and 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id="S2.p1.2.m2.1c">\hbar=2m=1</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">roman_ℏ = 2 italic_m = 1</annotation></semantics></math> and <math alttext="\Psi(x)" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><mrow id="S2.p1.3.m3.1.2" xref="S2.p1.3.m3.1.2.cmml"><mi id="S2.p1.3.m3.1.2.2" mathvariant="normal" xref="S2.p1.3.m3.1.2.2.cmml">Ψ</mi><mo id="S2.p1.3.m3.1.2.1" xref="S2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.p1.3.m3.1.2.3.2" xref="S2.p1.3.m3.1.2.cmml"><mo id="S2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S2.p1.3.m3.1.2.cmml">(</mo><mi id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml">x</mi><mo id="S2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><apply id="S2.p1.3.m3.1.2.cmml" xref="S2.p1.3.m3.1.2"><times id="S2.p1.3.m3.1.2.1.cmml" xref="S2.p1.3.m3.1.2.1"></times><ci id="S2.p1.3.m3.1.2.2.cmml" xref="S2.p1.3.m3.1.2.2">Ψ</ci><ci id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">\Psi(x)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">roman_Ψ ( italic_x )</annotation></semantics></math> is the wave function. The discrete version of the Schrödinger equation can be obtained by using the standard finite-difference scheme with the step size <math alttext="a>0" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><mrow id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><mi id="S2.p1.4.m4.1.1.2" xref="S2.p1.4.m4.1.1.2.cmml">a</mi><mo id="S2.p1.4.m4.1.1.1" xref="S2.p1.4.m4.1.1.1.cmml">></mo><mn id="S2.p1.4.m4.1.1.3" xref="S2.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.1b"><apply id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1"><gt id="S2.p1.4.m4.1.1.1.cmml" xref="S2.p1.4.m4.1.1.1"></gt><ci id="S2.p1.4.m4.1.1.2.cmml" xref="S2.p1.4.m4.1.1.2">𝑎</ci><cn id="S2.p1.4.m4.1.1.3.cmml" type="integer" xref="S2.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.1c">a>0</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.1d">italic_a > 0</annotation></semantics></math> for the second-order derivative as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx1"> <tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}\frac{1}{a^{2}}\bigg{[}\Psi^{(a)}(x_{n}^{(a)}-a)-2% \Psi^{(a)}(x_{n}^{(a)}&)+\Psi^{(a)}(x_{n}^{(a)}+a)\bigg{]}\\ &+k^{2}\Psi^{(a)}(x_{n}^{(a)})=0,\end{split}" class="ltx_Math" display="inline" id="S2.E2.m1.46"><semantics id="S2.E2.m1.46a"><mtable columnspacing="0pt" id="S2.E2.m1.46.46.2" rowspacing="0pt"><mtr id="S2.E2.m1.46.46.2a"><mtd class="ltx_align_right" columnalign="right" id="S2.E2.m1.46.46.2b"><mrow id="S2.E2.m1.19.19.19.19.19a"><mstyle displaystyle="true" id="S2.E2.m1.1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.1.cmml"><mfrac id="S2.E2.m1.1.1.1.1.1.1a" xref="S2.E2.m1.1.1.1.1.1.1.cmml"><mn id="S2.E2.m1.1.1.1.1.1.1.2" 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xref="S2.E2.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.E2.m1.45.45.1.1.1.2.2.1.1.1.2.1.cmml" xref="S2.E2.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S2.E2.m1.38.38.38.7.7.7.cmml" xref="S2.E2.m1.38.38.38.7.7.7">𝑥</ci><ci id="S2.E2.m1.39.39.39.8.8.8.1.cmml" xref="S2.E2.m1.39.39.39.8.8.8.1">𝑛</ci></apply><ci id="S2.E2.m1.40.40.40.9.9.9.1.1.cmml" xref="S2.E2.m1.40.40.40.9.9.9.1.1">𝑎</ci></apply></apply></apply><cn id="S2.E2.m1.43.43.43.12.12.12.cmml" type="integer" xref="S2.E2.m1.43.43.43.12.12.12">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.46c">\displaystyle\begin{split}\frac{1}{a^{2}}\bigg{[}\Psi^{(a)}(x_{n}^{(a)}-a)-2% \Psi^{(a)}(x_{n}^{(a)}&)+\Psi^{(a)}(x_{n}^{(a)}+a)\bigg{]}\\ &+k^{2}\Psi^{(a)}(x_{n}^{(a)})=0,\end{split}</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.46d">start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT - italic_a ) - 2 roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT end_CELL start_CELL ) + roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT + italic_a ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ) = 0 , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.6">where <math alttext="x_{n}^{(a)}=na" class="ltx_Math" display="inline" id="S2.p1.5.m1.1"><semantics id="S2.p1.5.m1.1a"><mrow id="S2.p1.5.m1.1.2" xref="S2.p1.5.m1.1.2.cmml"><msubsup id="S2.p1.5.m1.1.2.2" xref="S2.p1.5.m1.1.2.2.cmml"><mi id="S2.p1.5.m1.1.2.2.2.2" xref="S2.p1.5.m1.1.2.2.2.2.cmml">x</mi><mi id="S2.p1.5.m1.1.2.2.2.3" xref="S2.p1.5.m1.1.2.2.2.3.cmml">n</mi><mrow id="S2.p1.5.m1.1.1.1.3" xref="S2.p1.5.m1.1.2.2.cmml"><mo id="S2.p1.5.m1.1.1.1.3.1" stretchy="false" xref="S2.p1.5.m1.1.2.2.cmml">(</mo><mi id="S2.p1.5.m1.1.1.1.1" xref="S2.p1.5.m1.1.1.1.1.cmml">a</mi><mo id="S2.p1.5.m1.1.1.1.3.2" stretchy="false" xref="S2.p1.5.m1.1.2.2.cmml">)</mo></mrow></msubsup><mo id="S2.p1.5.m1.1.2.1" xref="S2.p1.5.m1.1.2.1.cmml">=</mo><mrow id="S2.p1.5.m1.1.2.3" xref="S2.p1.5.m1.1.2.3.cmml"><mi id="S2.p1.5.m1.1.2.3.2" xref="S2.p1.5.m1.1.2.3.2.cmml">n</mi><mo id="S2.p1.5.m1.1.2.3.1" xref="S2.p1.5.m1.1.2.3.1.cmml">⁢</mo><mi id="S2.p1.5.m1.1.2.3.3" xref="S2.p1.5.m1.1.2.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m1.1b"><apply id="S2.p1.5.m1.1.2.cmml" xref="S2.p1.5.m1.1.2"><eq id="S2.p1.5.m1.1.2.1.cmml" xref="S2.p1.5.m1.1.2.1"></eq><apply id="S2.p1.5.m1.1.2.2.cmml" xref="S2.p1.5.m1.1.2.2"><csymbol cd="ambiguous" id="S2.p1.5.m1.1.2.2.1.cmml" xref="S2.p1.5.m1.1.2.2">superscript</csymbol><apply id="S2.p1.5.m1.1.2.2.2.cmml" xref="S2.p1.5.m1.1.2.2"><csymbol cd="ambiguous" id="S2.p1.5.m1.1.2.2.2.1.cmml" xref="S2.p1.5.m1.1.2.2">subscript</csymbol><ci id="S2.p1.5.m1.1.2.2.2.2.cmml" xref="S2.p1.5.m1.1.2.2.2.2">𝑥</ci><ci id="S2.p1.5.m1.1.2.2.2.3.cmml" xref="S2.p1.5.m1.1.2.2.2.3">𝑛</ci></apply><ci id="S2.p1.5.m1.1.1.1.1.cmml" xref="S2.p1.5.m1.1.1.1.1">𝑎</ci></apply><apply id="S2.p1.5.m1.1.2.3.cmml" xref="S2.p1.5.m1.1.2.3"><times id="S2.p1.5.m1.1.2.3.1.cmml" xref="S2.p1.5.m1.1.2.3.1"></times><ci id="S2.p1.5.m1.1.2.3.2.cmml" xref="S2.p1.5.m1.1.2.3.2">𝑛</ci><ci id="S2.p1.5.m1.1.2.3.3.cmml" xref="S2.p1.5.m1.1.2.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m1.1c">x_{n}^{(a)}=na</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m1.1d">italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT = italic_n italic_a</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S2.p1.6.m2.1"><semantics id="S2.p1.6.m2.1a"><mi id="S2.p1.6.m2.1.1" xref="S2.p1.6.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m2.1b"><ci id="S2.p1.6.m2.1.1.cmml" xref="S2.p1.6.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m2.1d">italic_n</annotation></semantics></math> is an integer number.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.3">Here we propose an exact solution of Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E2" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a>) which can be written as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Psi^{(a)}(x_{n}^{(a)})=Ag_{+}(a)^{x_{n}^{(a)}/a}+Bg_{-}(a)^{x_{n}^{(a)}/a}," class="ltx_Math" display="block" id="S2.E3.m1.7"><semantics id="S2.E3.m1.7a"><mrow id="S2.E3.m1.7.7.1" xref="S2.E3.m1.7.7.1.1.cmml"><mrow id="S2.E3.m1.7.7.1.1" xref="S2.E3.m1.7.7.1.1.cmml"><mrow id="S2.E3.m1.7.7.1.1.1" xref="S2.E3.m1.7.7.1.1.1.cmml"><msup id="S2.E3.m1.7.7.1.1.1.3" xref="S2.E3.m1.7.7.1.1.1.3.cmml"><mi id="S2.E3.m1.7.7.1.1.1.3.2" 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italic_a ) end_POSTSUPERSCRIPT / italic_a end_POSTSUPERSCRIPT + italic_B italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT / italic_a end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p2.2">where <math alttext="A" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mi id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><ci id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">italic_B</annotation></semantics></math> are constants and</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{\pm}(a)=1+\frac{-k^{2}a^{2}\pm ka\sqrt{k^{2}a^{2}-4}}{2}." class="ltx_Math" display="block" id="S2.Ex1.m1.2"><semantics id="S2.Ex1.m1.2a"><mrow id="S2.Ex1.m1.2.2.1" 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xref="S2.Ex1.m1.2.2.1.1.3.3.2.3.4.2.2.3">superscript</csymbol><ci id="S2.Ex1.m1.2.2.1.1.3.3.2.3.4.2.2.3.2.cmml" xref="S2.Ex1.m1.2.2.1.1.3.3.2.3.4.2.2.3.2">𝑎</ci><cn id="S2.Ex1.m1.2.2.1.1.3.3.2.3.4.2.2.3.3.cmml" type="integer" xref="S2.Ex1.m1.2.2.1.1.3.3.2.3.4.2.2.3.3">2</cn></apply></apply><cn id="S2.Ex1.m1.2.2.1.1.3.3.2.3.4.2.3.cmml" type="integer" xref="S2.Ex1.m1.2.2.1.1.3.3.2.3.4.2.3">4</cn></apply></apply></apply></apply><cn id="S2.Ex1.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S2.Ex1.m1.2.2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.2c">g_{\pm}(a)=1+\frac{-k^{2}a^{2}\pm ka\sqrt{k^{2}a^{2}-4}}{2}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.2d">italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) = 1 + divide start_ARG - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ± italic_k italic_a square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 4 end_ARG end_ARG start_ARG 2 end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <figure class="ltx_figure" id="S2.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="327" id="S2.F1.g1" src="x1.png" width="463"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Analytic solution of discrete Schrödinger equation in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E3" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">3</span></a>) for different values of the step size <math alttext="a" class="ltx_Math" display="inline" id="S2.F1.4.m1.1"><semantics id="S2.F1.4.m1.1b"><mi id="S2.F1.4.m1.1.1" xref="S2.F1.4.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.F1.4.m1.1c"><ci id="S2.F1.4.m1.1.1.cmml" xref="S2.F1.4.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.4.m1.1d">a</annotation><annotation encoding="application/x-llamapun" id="S2.F1.4.m1.1e">italic_a</annotation></semantics></math> and limit of the solution in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E5" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">5</span></a>) for parameters <math alttext="k=0.8" class="ltx_Math" display="inline" id="S2.F1.5.m2.1"><semantics id="S2.F1.5.m2.1b"><mrow id="S2.F1.5.m2.1.1" xref="S2.F1.5.m2.1.1.cmml"><mi id="S2.F1.5.m2.1.1.2" xref="S2.F1.5.m2.1.1.2.cmml">k</mi><mo id="S2.F1.5.m2.1.1.1" xref="S2.F1.5.m2.1.1.1.cmml">=</mo><mn id="S2.F1.5.m2.1.1.3" xref="S2.F1.5.m2.1.1.3.cmml">0.8</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.5.m2.1c"><apply id="S2.F1.5.m2.1.1.cmml" xref="S2.F1.5.m2.1.1"><eq id="S2.F1.5.m2.1.1.1.cmml" xref="S2.F1.5.m2.1.1.1"></eq><ci id="S2.F1.5.m2.1.1.2.cmml" xref="S2.F1.5.m2.1.1.2">𝑘</ci><cn id="S2.F1.5.m2.1.1.3.cmml" type="float" xref="S2.F1.5.m2.1.1.3">0.8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.5.m2.1d">k=0.8</annotation><annotation encoding="application/x-llamapun" id="S2.F1.5.m2.1e">italic_k = 0.8</annotation></semantics></math>, <math alttext="A=B=1" class="ltx_Math" display="inline" id="S2.F1.6.m3.1"><semantics id="S2.F1.6.m3.1b"><mrow id="S2.F1.6.m3.1.1" xref="S2.F1.6.m3.1.1.cmml"><mi id="S2.F1.6.m3.1.1.2" xref="S2.F1.6.m3.1.1.2.cmml">A</mi><mo id="S2.F1.6.m3.1.1.3" xref="S2.F1.6.m3.1.1.3.cmml">=</mo><mi id="S2.F1.6.m3.1.1.4" xref="S2.F1.6.m3.1.1.4.cmml">B</mi><mo id="S2.F1.6.m3.1.1.5" xref="S2.F1.6.m3.1.1.5.cmml">=</mo><mn id="S2.F1.6.m3.1.1.6" xref="S2.F1.6.m3.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.6.m3.1c"><apply id="S2.F1.6.m3.1.1.cmml" xref="S2.F1.6.m3.1.1"><and id="S2.F1.6.m3.1.1a.cmml" xref="S2.F1.6.m3.1.1"></and><apply id="S2.F1.6.m3.1.1b.cmml" xref="S2.F1.6.m3.1.1"><eq id="S2.F1.6.m3.1.1.3.cmml" xref="S2.F1.6.m3.1.1.3"></eq><ci id="S2.F1.6.m3.1.1.2.cmml" xref="S2.F1.6.m3.1.1.2">𝐴</ci><ci id="S2.F1.6.m3.1.1.4.cmml" xref="S2.F1.6.m3.1.1.4">𝐵</ci></apply><apply id="S2.F1.6.m3.1.1c.cmml" xref="S2.F1.6.m3.1.1"><eq id="S2.F1.6.m3.1.1.5.cmml" xref="S2.F1.6.m3.1.1.5"></eq><share href="https://arxiv.org/html/2411.14397v1#S2.F1.6.m3.1.1.4.cmml" id="S2.F1.6.m3.1.1d.cmml" xref="S2.F1.6.m3.1.1"></share><cn id="S2.F1.6.m3.1.1.6.cmml" type="integer" xref="S2.F1.6.m3.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.6.m3.1d">A=B=1</annotation><annotation encoding="application/x-llamapun" id="S2.F1.6.m3.1e">italic_A = italic_B = 1</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.2">The consistency of this solution with the solution of the continuous case can be shown by considering its limit for <math alttext="a\to 0" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><mi id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml">a</mi><mo id="S2.p3.1.m1.1.1.1" stretchy="false" xref="S2.p3.1.m1.1.1.1.cmml">→</mo><mn id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><ci id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1.1">→</ci><ci id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2">𝑎</ci><cn id="S2.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">a\to 0</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_a → 0</annotation></semantics></math> for a fixed point <math alttext="x_{n}^{(a)}" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><msubsup id="S2.p3.2.m2.1.2" xref="S2.p3.2.m2.1.2.cmml"><mi id="S2.p3.2.m2.1.2.2.2" xref="S2.p3.2.m2.1.2.2.2.cmml">x</mi><mi id="S2.p3.2.m2.1.2.2.3" xref="S2.p3.2.m2.1.2.2.3.cmml">n</mi><mrow id="S2.p3.2.m2.1.1.1.3" xref="S2.p3.2.m2.1.2.cmml"><mo id="S2.p3.2.m2.1.1.1.3.1" stretchy="false" xref="S2.p3.2.m2.1.2.cmml">(</mo><mi id="S2.p3.2.m2.1.1.1.1" xref="S2.p3.2.m2.1.1.1.1.cmml">a</mi><mo id="S2.p3.2.m2.1.1.1.3.2" stretchy="false" xref="S2.p3.2.m2.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.2.cmml" xref="S2.p3.2.m2.1.2"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.2.1.cmml" xref="S2.p3.2.m2.1.2">superscript</csymbol><apply id="S2.p3.2.m2.1.2.2.cmml" xref="S2.p3.2.m2.1.2"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.2.2.1.cmml" xref="S2.p3.2.m2.1.2">subscript</csymbol><ci id="S2.p3.2.m2.1.2.2.2.cmml" xref="S2.p3.2.m2.1.2.2.2">𝑥</ci><ci id="S2.p3.2.m2.1.2.2.3.cmml" xref="S2.p3.2.m2.1.2.2.3">𝑛</ci></apply><ci id="S2.p3.2.m2.1.1.1.1.cmml" xref="S2.p3.2.m2.1.1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">x_{n}^{(a)}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT</annotation></semantics></math>. For brevity we set</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="z_{\pm}:=\frac{-k^{2}a^{2}\pm ka\sqrt{k^{2}a^{2}-4}}{2}\quad\mbox{and}\quad% \alpha:=x_{n}^{(a)}." class="ltx_Math" display="block" id="S2.Ex2.m1.4"><semantics id="S2.Ex2.m1.4a"><mrow id="S2.Ex2.m1.4.4.1"><mrow id="S2.Ex2.m1.4.4.1.1.2" xref="S2.Ex2.m1.4.4.1.1.3.cmml"><mrow id="S2.Ex2.m1.4.4.1.1.1.1" xref="S2.Ex2.m1.4.4.1.1.1.1.cmml"><msub id="S2.Ex2.m1.4.4.1.1.1.1.2" xref="S2.Ex2.m1.4.4.1.1.1.1.2.cmml"><mi id="S2.Ex2.m1.4.4.1.1.1.1.2.2" xref="S2.Ex2.m1.4.4.1.1.1.1.2.2.cmml">z</mi><mo id="S2.Ex2.m1.4.4.1.1.1.1.2.3" xref="S2.Ex2.m1.4.4.1.1.1.1.2.3.cmml">±</mo></msub><mo id="S2.Ex2.m1.4.4.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex2.m1.4.4.1.1.1.1.1.cmml">:=</mo><mrow id="S2.Ex2.m1.4.4.1.1.1.1.3.2" xref="S2.Ex2.m1.4.4.1.1.1.1.3.1.cmml"><mfrac id="S2.Ex2.m1.2.2" xref="S2.Ex2.m1.2.2.cmml"><mrow id="S2.Ex2.m1.2.2.2" xref="S2.Ex2.m1.2.2.2.cmml"><mrow id="S2.Ex2.m1.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.cmml"><mo id="S2.Ex2.m1.2.2.2.2a" xref="S2.Ex2.m1.2.2.2.2.cmml">−</mo><mrow id="S2.Ex2.m1.2.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.2.cmml"><msup id="S2.Ex2.m1.2.2.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.2.2.cmml"><mi id="S2.Ex2.m1.2.2.2.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.2.2.2.cmml">k</mi><mn id="S2.Ex2.m1.2.2.2.2.2.2.3" xref="S2.Ex2.m1.2.2.2.2.2.2.3.cmml">2</mn></msup><mo id="S2.Ex2.m1.2.2.2.2.2.1" xref="S2.Ex2.m1.2.2.2.2.2.1.cmml">⁢</mo><msup id="S2.Ex2.m1.2.2.2.2.2.3" xref="S2.Ex2.m1.2.2.2.2.2.3.cmml"><mi id="S2.Ex2.m1.2.2.2.2.2.3.2" xref="S2.Ex2.m1.2.2.2.2.2.3.2.cmml">a</mi><mn id="S2.Ex2.m1.2.2.2.2.2.3.3" xref="S2.Ex2.m1.2.2.2.2.2.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S2.Ex2.m1.2.2.2.1" xref="S2.Ex2.m1.2.2.2.1.cmml">±</mo><mrow id="S2.Ex2.m1.2.2.2.3" xref="S2.Ex2.m1.2.2.2.3.cmml"><mi id="S2.Ex2.m1.2.2.2.3.2" xref="S2.Ex2.m1.2.2.2.3.2.cmml">k</mi><mo id="S2.Ex2.m1.2.2.2.3.1" xref="S2.Ex2.m1.2.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex2.m1.2.2.2.3.3" xref="S2.Ex2.m1.2.2.2.3.3.cmml">a</mi><mo id="S2.Ex2.m1.2.2.2.3.1a" xref="S2.Ex2.m1.2.2.2.3.1.cmml">⁢</mo><msqrt id="S2.Ex2.m1.2.2.2.3.4" xref="S2.Ex2.m1.2.2.2.3.4.cmml"><mrow id="S2.Ex2.m1.2.2.2.3.4.2" xref="S2.Ex2.m1.2.2.2.3.4.2.cmml"><mrow id="S2.Ex2.m1.2.2.2.3.4.2.2" xref="S2.Ex2.m1.2.2.2.3.4.2.2.cmml"><msup id="S2.Ex2.m1.2.2.2.3.4.2.2.2" xref="S2.Ex2.m1.2.2.2.3.4.2.2.2.cmml"><mi id="S2.Ex2.m1.2.2.2.3.4.2.2.2.2" xref="S2.Ex2.m1.2.2.2.3.4.2.2.2.2.cmml">k</mi><mn id="S2.Ex2.m1.2.2.2.3.4.2.2.2.3" xref="S2.Ex2.m1.2.2.2.3.4.2.2.2.3.cmml">2</mn></msup><mo id="S2.Ex2.m1.2.2.2.3.4.2.2.1" xref="S2.Ex2.m1.2.2.2.3.4.2.2.1.cmml">⁢</mo><msup id="S2.Ex2.m1.2.2.2.3.4.2.2.3" xref="S2.Ex2.m1.2.2.2.3.4.2.2.3.cmml"><mi id="S2.Ex2.m1.2.2.2.3.4.2.2.3.2" xref="S2.Ex2.m1.2.2.2.3.4.2.2.3.2.cmml">a</mi><mn id="S2.Ex2.m1.2.2.2.3.4.2.2.3.3" 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start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p3.4">As <math alttext="z_{\pm}" class="ltx_Math" display="inline" id="S2.p3.3.m1.1"><semantics id="S2.p3.3.m1.1a"><msub id="S2.p3.3.m1.1.1" xref="S2.p3.3.m1.1.1.cmml"><mi id="S2.p3.3.m1.1.1.2" xref="S2.p3.3.m1.1.1.2.cmml">z</mi><mo id="S2.p3.3.m1.1.1.3" xref="S2.p3.3.m1.1.1.3.cmml">±</mo></msub><annotation-xml encoding="MathML-Content" id="S2.p3.3.m1.1b"><apply id="S2.p3.3.m1.1.1.cmml" xref="S2.p3.3.m1.1.1"><csymbol cd="ambiguous" id="S2.p3.3.m1.1.1.1.cmml" xref="S2.p3.3.m1.1.1">subscript</csymbol><ci id="S2.p3.3.m1.1.1.2.cmml" xref="S2.p3.3.m1.1.1.2">𝑧</ci><csymbol cd="latexml" id="S2.p3.3.m1.1.1.3.cmml" xref="S2.p3.3.m1.1.1.3">plus-or-minus</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m1.1c">z_{\pm}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m1.1d">italic_z start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT</annotation></semantics></math> tends to zero for <math alttext="a\to 0" class="ltx_Math" display="inline" id="S2.p3.4.m2.1"><semantics id="S2.p3.4.m2.1a"><mrow id="S2.p3.4.m2.1.1" xref="S2.p3.4.m2.1.1.cmml"><mi id="S2.p3.4.m2.1.1.2" xref="S2.p3.4.m2.1.1.2.cmml">a</mi><mo id="S2.p3.4.m2.1.1.1" stretchy="false" xref="S2.p3.4.m2.1.1.1.cmml">→</mo><mn id="S2.p3.4.m2.1.1.3" xref="S2.p3.4.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m2.1b"><apply id="S2.p3.4.m2.1.1.cmml" xref="S2.p3.4.m2.1.1"><ci id="S2.p3.4.m2.1.1.1.cmml" xref="S2.p3.4.m2.1.1.1">→</ci><ci id="S2.p3.4.m2.1.1.2.cmml" xref="S2.p3.4.m2.1.1.2">𝑎</ci><cn id="S2.p3.4.m2.1.1.3.cmml" type="integer" xref="S2.p3.4.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m2.1c">a\to 0</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m2.1d">italic_a → 0</annotation></semantics></math>, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx2"> <tbody id="S2.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}\lim_{a\to 0}\left(g_{\pm}(a)\right)^{\alpha/a}&=% \lim_{a\to 0}\left(1+z_{\pm}\right)^{\alpha/a}\\ &=\lim_{a\to 0}\left[\left(1+z_{\pm}\right)^{1/z_{\pm}}\right]^{\alpha z_{\pm}% /a}\\ &=\exp{\left(\lim_{a\to 0}\alpha z_{\pm}/a\right)}=e^{\pm ik\alpha}.\end{split}" class="ltx_Math" display="inline" id="S2.E4.m1.53"><semantics id="S2.E4.m1.53a"><mtable columnspacing="0pt" id="S2.E4.m1.53.53.5" rowspacing="0pt"><mtr id="S2.E4.m1.53.53.5a"><mtd class="ltx_align_right" columnalign="right" id="S2.E4.m1.53.53.5b"><mrow id="S2.E4.m1.50.50.2.49.21.11"><munder 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xref="S2.E4.m1.18.18.18.18.8.8.1">plus-or-minus</csymbol></apply></apply><apply id="S2.E4.m1.20.20.20.20.10.10.1.cmml" xref="S2.E4.m1.20.20.20.20.10.10.1"><divide id="S2.E4.m1.20.20.20.20.10.10.1.1.cmml" xref="S2.E4.m1.20.20.20.20.10.10.1.1"></divide><ci id="S2.E4.m1.20.20.20.20.10.10.1.2.cmml" xref="S2.E4.m1.20.20.20.20.10.10.1.2">𝛼</ci><ci id="S2.E4.m1.20.20.20.20.10.10.1.3.cmml" xref="S2.E4.m1.20.20.20.20.10.10.1.3">𝑎</ci></apply></apply></apply></apply><apply id="S2.E4.m1.49.49.1.1.1c.cmml" xref="S2.E4.m1.3.3.3.3.3.3"><eq id="S2.E4.m1.21.21.21.1.1.1.cmml" xref="S2.E4.m1.21.21.21.1.1.1"></eq><share href="https://arxiv.org/html/2411.14397v1#S2.E4.m1.49.49.1.1.1.2.cmml" id="S2.E4.m1.49.49.1.1.1d.cmml" xref="S2.E4.m1.3.3.3.3.3.3"></share><apply id="S2.E4.m1.49.49.1.1.1.3.cmml" xref="S2.E4.m1.3.3.3.3.3.3"><apply id="S2.E4.m1.49.49.1.1.1.3.2.cmml" xref="S2.E4.m1.3.3.3.3.3.3"><csymbol cd="ambiguous" id="S2.E4.m1.49.49.1.1.1.3.2.1.cmml" xref="S2.E4.m1.3.3.3.3.3.3">subscript</csymbol><limit id="S2.E4.m1.22.22.22.2.2.2.cmml" xref="S2.E4.m1.22.22.22.2.2.2"></limit><apply id="S2.E4.m1.23.23.23.3.3.3.1.cmml" xref="S2.E4.m1.23.23.23.3.3.3.1"><ci id="S2.E4.m1.23.23.23.3.3.3.1.1.cmml" xref="S2.E4.m1.23.23.23.3.3.3.1.1">→</ci><ci id="S2.E4.m1.23.23.23.3.3.3.1.2.cmml" xref="S2.E4.m1.23.23.23.3.3.3.1.2">𝑎</ci><cn id="S2.E4.m1.23.23.23.3.3.3.1.3.cmml" type="integer" xref="S2.E4.m1.23.23.23.3.3.3.1.3">0</cn></apply></apply><apply id="S2.E4.m1.49.49.1.1.1.3.1.cmml" xref="S2.E4.m1.3.3.3.3.3.3"><csymbol cd="ambiguous" id="S2.E4.m1.49.49.1.1.1.3.1.2.cmml" xref="S2.E4.m1.3.3.3.3.3.3">superscript</csymbol><apply id="S2.E4.m1.49.49.1.1.1.3.1.1.2.cmml" xref="S2.E4.m1.3.3.3.3.3.3"><csymbol cd="latexml" id="S2.E4.m1.49.49.1.1.1.3.1.1.2.1.cmml" xref="S2.E4.m1.3.3.3.3.3.3">delimited-[]</csymbol><apply id="S2.E4.m1.49.49.1.1.1.3.1.1.1.1.cmml" xref="S2.E4.m1.3.3.3.3.3.3"><csymbol cd="ambiguous" id="S2.E4.m1.49.49.1.1.1.3.1.1.1.1.2.cmml" xref="S2.E4.m1.3.3.3.3.3.3">superscript</csymbol><apply 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xref="S2.E4.m1.31.31.31.11.11.11.1.3"><csymbol cd="ambiguous" id="S2.E4.m1.31.31.31.11.11.11.1.3.1.cmml" xref="S2.E4.m1.31.31.31.11.11.11.1.3">subscript</csymbol><ci id="S2.E4.m1.31.31.31.11.11.11.1.3.2.cmml" xref="S2.E4.m1.31.31.31.11.11.11.1.3.2">𝑧</ci><csymbol cd="latexml" id="S2.E4.m1.31.31.31.11.11.11.1.3.3.cmml" xref="S2.E4.m1.31.31.31.11.11.11.1.3.3">plus-or-minus</csymbol></apply></apply></apply></apply><apply id="S2.E4.m1.33.33.33.13.13.13.1.cmml" xref="S2.E4.m1.33.33.33.13.13.13.1"><divide id="S2.E4.m1.33.33.33.13.13.13.1.1.cmml" xref="S2.E4.m1.33.33.33.13.13.13.1.1"></divide><apply id="S2.E4.m1.33.33.33.13.13.13.1.2.cmml" xref="S2.E4.m1.33.33.33.13.13.13.1.2"><times id="S2.E4.m1.33.33.33.13.13.13.1.2.1.cmml" xref="S2.E4.m1.33.33.33.13.13.13.1.2.1"></times><ci id="S2.E4.m1.33.33.33.13.13.13.1.2.2.cmml" xref="S2.E4.m1.33.33.33.13.13.13.1.2.2">𝛼</ci><apply id="S2.E4.m1.33.33.33.13.13.13.1.2.3.cmml" xref="S2.E4.m1.33.33.33.13.13.13.1.2.3"><csymbol cd="ambiguous" 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xref="S2.E4.m1.3.3.3.3.3.3">superscript</csymbol><ci id="S2.E4.m1.46.46.46.13.13.13.cmml" xref="S2.E4.m1.46.46.46.13.13.13">𝑒</ci><apply id="S2.E4.m1.47.47.47.14.14.14.1.cmml" xref="S2.E4.m1.47.47.47.14.14.14.1"><csymbol cd="latexml" id="S2.E4.m1.47.47.47.14.14.14.1.1.cmml" xref="S2.E4.m1.47.47.47.14.14.14.1">plus-or-minus</csymbol><apply id="S2.E4.m1.47.47.47.14.14.14.1.2.cmml" xref="S2.E4.m1.47.47.47.14.14.14.1.2"><times id="S2.E4.m1.47.47.47.14.14.14.1.2.1.cmml" xref="S2.E4.m1.47.47.47.14.14.14.1.2.1"></times><ci id="S2.E4.m1.47.47.47.14.14.14.1.2.2.cmml" xref="S2.E4.m1.47.47.47.14.14.14.1.2.2">𝑖</ci><ci id="S2.E4.m1.47.47.47.14.14.14.1.2.3.cmml" xref="S2.E4.m1.47.47.47.14.14.14.1.2.3">𝑘</ci><ci id="S2.E4.m1.47.47.47.14.14.14.1.2.4.cmml" xref="S2.E4.m1.47.47.47.14.14.14.1.2.4">𝛼</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.53c">\displaystyle\begin{split}\lim_{a\to 0}\left(g_{\pm}(a)\right)^{\alpha/a}&=% \lim_{a\to 0}\left(1+z_{\pm}\right)^{\alpha/a}\\ &=\lim_{a\to 0}\left[\left(1+z_{\pm}\right)^{1/z_{\pm}}\right]^{\alpha z_{\pm}% /a}\\ &=\exp{\left(\lim_{a\to 0}\alpha z_{\pm}/a\right)}=e^{\pm ik\alpha}.\end{split}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.53d">start_ROW start_CELL roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) ) start_POSTSUPERSCRIPT italic_α / italic_a end_POSTSUPERSCRIPT end_CELL start_CELL = roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT ( 1 + italic_z start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_α / italic_a end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT [ ( 1 + italic_z start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 1 / italic_z start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT italic_α italic_z start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT / italic_a end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = roman_exp ( roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT italic_α italic_z start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT / italic_a ) = italic_e start_POSTSUPERSCRIPT ± italic_i italic_k italic_α end_POSTSUPERSCRIPT . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p3.6">One can easily show that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx3"> <tbody id="S2.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}\lim_{a\to 0}\Psi^{(a)}(\alpha)=\lim_{a\to 0}\bigg{(% }A[g_{+}(a)]&{}^{\alpha/a}+B[g_{-}(a)]^{\alpha/a}\bigg{)}\\ &=Ae^{ik\alpha}+Be^{-ik\alpha}.\end{split}" class="ltx_math_unparsed" display="inline" id="S2.E5.m1.41"><semantics id="S2.E5.m1.41a"><mtable columnspacing="0pt" id="S2.E5.m1.41.41.1" rowspacing="0pt"><mtr id="S2.E5.m1.41.41.1a"><mtd class="ltx_align_right" columnalign="right" id="S2.E5.m1.41.41.1b"><mrow id="S2.E5.m1.19.19.19.19.19a"><munder id="S2.E5.m1.19.19.19.19.19a.20"><mo id="S2.E5.m1.1.1.1.1.1.1" movablelimits="false">lim</mo><mrow id="S2.E5.m1.2.2.2.2.2.2.1"><mi id="S2.E5.m1.2.2.2.2.2.2.1.2">a</mi><mo id="S2.E5.m1.2.2.2.2.2.2.1.1" stretchy="false">→</mo><mn id="S2.E5.m1.2.2.2.2.2.2.1.3">0</mn></mrow></munder><msup id="S2.E5.m1.19.19.19.19.19a.21"><mi id="S2.E5.m1.3.3.3.3.3.3" mathvariant="normal">Ψ</mi><mrow id="S2.E5.m1.4.4.4.4.4.4.1.3"><mo id="S2.E5.m1.4.4.4.4.4.4.1.3.1" stretchy="false">(</mo><mi id="S2.E5.m1.4.4.4.4.4.4.1.1">a</mi><mo id="S2.E5.m1.4.4.4.4.4.4.1.3.2" stretchy="false">)</mo></mrow></msup><mrow id="S2.E5.m1.19.19.19.19.19a.22"><mo id="S2.E5.m1.5.5.5.5.5.5" stretchy="false">(</mo><mi id="S2.E5.m1.6.6.6.6.6.6">α</mi><mo id="S2.E5.m1.7.7.7.7.7.7" stretchy="false">)</mo></mrow><mo id="S2.E5.m1.8.8.8.8.8.8" rspace="0.1389em">=</mo><munder id="S2.E5.m1.19.19.19.19.19a.23"><mo id="S2.E5.m1.9.9.9.9.9.9" lspace="0.1389em" movablelimits="false" rspace="0em">lim</mo><mrow id="S2.E5.m1.10.10.10.10.10.10.1"><mi id="S2.E5.m1.10.10.10.10.10.10.1.2">a</mi><mo id="S2.E5.m1.10.10.10.10.10.10.1.1" stretchy="false">→</mo><mn id="S2.E5.m1.10.10.10.10.10.10.1.3">0</mn></mrow></munder><mrow id="S2.E5.m1.19.19.19.19.19a.24"><mo id="S2.E5.m1.11.11.11.11.11.11" maxsize="210%" minsize="210%">(</mo><mi id="S2.E5.m1.12.12.12.12.12.12">A</mi><mrow id="S2.E5.m1.19.19.19.19.19a.24.1"><mo id="S2.E5.m1.13.13.13.13.13.13" stretchy="false">[</mo><msub id="S2.E5.m1.19.19.19.19.19a.24.1.1"><mi id="S2.E5.m1.14.14.14.14.14.14">g</mi><mo id="S2.E5.m1.15.15.15.15.15.15.1">+</mo></msub><mrow id="S2.E5.m1.19.19.19.19.19a.24.1.2"><mo id="S2.E5.m1.16.16.16.16.16.16" stretchy="false">(</mo><mi id="S2.E5.m1.17.17.17.17.17.17">a</mi><mo id="S2.E5.m1.18.18.18.18.18.18" stretchy="false">)</mo></mrow><mo id="S2.E5.m1.19.19.19.19.19.19" stretchy="false">]</mo></mrow></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.E5.m1.41.41.1c"><mrow id="S2.E5.m1.31.31.31.31.12a"><mmultiscripts id="S2.E5.m1.31.31.31.31.12a.13"><mo id="S2.E5.m1.21.21.21.21.2.2">+</mo><mprescripts id="S2.E5.m1.31.31.31.31.12a.13a"></mprescripts><mrow id="S2.E5.m1.31.31.31.31.12a.13b"></mrow><mrow id="S2.E5.m1.20.20.20.20.1.1.1"><mi id="S2.E5.m1.20.20.20.20.1.1.1.2">α</mi><mo id="S2.E5.m1.20.20.20.20.1.1.1.1">/</mo><mi id="S2.E5.m1.20.20.20.20.1.1.1.3">a</mi></mrow></mmultiscripts><mi id="S2.E5.m1.22.22.22.22.3.3">B</mi><msup id="S2.E5.m1.31.31.31.31.12a.14"><mrow id="S2.E5.m1.31.31.31.31.12a.14.2"><mo id="S2.E5.m1.23.23.23.23.4.4" stretchy="false">[</mo><msub id="S2.E5.m1.31.31.31.31.12a.14.2.1"><mi id="S2.E5.m1.24.24.24.24.5.5">g</mi><mo id="S2.E5.m1.25.25.25.25.6.6.1">−</mo></msub><mrow id="S2.E5.m1.31.31.31.31.12a.14.2.2"><mo id="S2.E5.m1.26.26.26.26.7.7" stretchy="false">(</mo><mi id="S2.E5.m1.27.27.27.27.8.8">a</mi><mo id="S2.E5.m1.28.28.28.28.9.9" stretchy="false">)</mo></mrow><mo id="S2.E5.m1.29.29.29.29.10.10" stretchy="false">]</mo></mrow><mrow id="S2.E5.m1.30.30.30.30.11.11.1"><mi id="S2.E5.m1.30.30.30.30.11.11.1.2">α</mi><mo id="S2.E5.m1.30.30.30.30.11.11.1.1">/</mo><mi id="S2.E5.m1.30.30.30.30.11.11.1.3">a</mi></mrow></msup><mo id="S2.E5.m1.31.31.31.31.12.12" maxsize="210%" minsize="210%">)</mo></mrow></mtd></mtr><mtr id="S2.E5.m1.41.41.1d"><mtd id="S2.E5.m1.41.41.1e"></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.E5.m1.41.41.1f"><mrow id="S2.E5.m1.41.41.1.41.10.10.10"><mrow id="S2.E5.m1.41.41.1.41.10.10.10.1"><mi id="S2.E5.m1.41.41.1.41.10.10.10.1.1"></mi><mo id="S2.E5.m1.32.32.32.1.1.1">=</mo><mrow id="S2.E5.m1.41.41.1.41.10.10.10.1.2"><mrow id="S2.E5.m1.41.41.1.41.10.10.10.1.2.1"><mi id="S2.E5.m1.33.33.33.2.2.2">A</mi><mo id="S2.E5.m1.41.41.1.41.10.10.10.1.2.1.1">⁢</mo><msup id="S2.E5.m1.41.41.1.41.10.10.10.1.2.1.2"><mi id="S2.E5.m1.34.34.34.3.3.3">e</mi><mrow id="S2.E5.m1.35.35.35.4.4.4.1"><mi id="S2.E5.m1.35.35.35.4.4.4.1.2">i</mi><mo id="S2.E5.m1.35.35.35.4.4.4.1.1">⁢</mo><mi id="S2.E5.m1.35.35.35.4.4.4.1.3">k</mi><mo id="S2.E5.m1.35.35.35.4.4.4.1.1a">⁢</mo><mi id="S2.E5.m1.35.35.35.4.4.4.1.4">α</mi></mrow></msup></mrow><mo id="S2.E5.m1.36.36.36.5.5.5">+</mo><mrow id="S2.E5.m1.41.41.1.41.10.10.10.1.2.2"><mi id="S2.E5.m1.37.37.37.6.6.6">B</mi><mo id="S2.E5.m1.41.41.1.41.10.10.10.1.2.2.1">⁢</mo><msup id="S2.E5.m1.41.41.1.41.10.10.10.1.2.2.2"><mi id="S2.E5.m1.38.38.38.7.7.7">e</mi><mrow id="S2.E5.m1.39.39.39.8.8.8.1"><mo id="S2.E5.m1.39.39.39.8.8.8.1a">−</mo><mrow id="S2.E5.m1.39.39.39.8.8.8.1.2"><mi id="S2.E5.m1.39.39.39.8.8.8.1.2.2">i</mi><mo id="S2.E5.m1.39.39.39.8.8.8.1.2.1">⁢</mo><mi id="S2.E5.m1.39.39.39.8.8.8.1.2.3">k</mi><mo id="S2.E5.m1.39.39.39.8.8.8.1.2.1a">⁢</mo><mi id="S2.E5.m1.39.39.39.8.8.8.1.2.4">α</mi></mrow></mrow></msup></mrow></mrow></mrow><mo id="S2.E5.m1.40.40.40.9.9.9" lspace="0em">.</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.E5.m1.41b">\displaystyle\begin{split}\lim_{a\to 0}\Psi^{(a)}(\alpha)=\lim_{a\to 0}\bigg{(% }A[g_{+}(a)]&{}^{\alpha/a}+B[g_{-}(a)]^{\alpha/a}\bigg{)}\\ &=Ae^{ik\alpha}+Be^{-ik\alpha}.\end{split}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.41c">start_ROW start_CELL roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_α ) = roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT ( italic_A [ italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) ] end_CELL start_CELL start_FLOATSUPERSCRIPT italic_α / italic_a end_FLOATSUPERSCRIPT + italic_B [ italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) ] start_POSTSUPERSCRIPT italic_α / italic_a end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = italic_A italic_e start_POSTSUPERSCRIPT italic_i italic_k italic_α end_POSTSUPERSCRIPT + italic_B italic_e start_POSTSUPERSCRIPT - italic_i italic_k italic_α end_POSTSUPERSCRIPT . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p3.5">Graphical representation of the convergence of the solution (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E3" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">3</span></a>) to the solution of the continuous case with a decrease of the step size <math alttext="a" class="ltx_Math" display="inline" id="S2.p3.5.m1.1"><semantics id="S2.p3.5.m1.1a"><mi id="S2.p3.5.m1.1.1" xref="S2.p3.5.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.p3.5.m1.1b"><ci id="S2.p3.5.m1.1.1.cmml" xref="S2.p3.5.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m1.1d">italic_a</annotation></semantics></math> is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.F1" title="Figure 1 ‣ 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <figure class="ltx_table" id="S2.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 1: </span>The first five nonzero eigenvalues <math alttext="k" class="ltx_Math" display="inline" id="S2.T1.2.m1.1"><semantics id="S2.T1.2.m1.1b"><mi id="S2.T1.2.m1.1.1" xref="S2.T1.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.T1.2.m1.1c"><ci id="S2.T1.2.m1.1.1.cmml" xref="S2.T1.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.2.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="S2.T1.2.m1.1e">italic_k</annotation></semantics></math> for Dirichlet (left) and Neumann (right) boundary condition compared with the continuous case.</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <table class="ltx_tabular ltx_figure_panel ltx_parbox ltx_align_middle" id="S2.T1.13.11" style="width:212.5pt;"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S2.T1.5.3.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r" id="S2.T1.5.3.3.4">Continuous</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.3.1.1.1"><math alttext="a=0.1" class="ltx_Math" display="inline" id="S2.T1.3.1.1.1.m1.1"><semantics id="S2.T1.3.1.1.1.m1.1a"><mrow id="S2.T1.3.1.1.1.m1.1.1" xref="S2.T1.3.1.1.1.m1.1.1.cmml"><mi id="S2.T1.3.1.1.1.m1.1.1.2" xref="S2.T1.3.1.1.1.m1.1.1.2.cmml">a</mi><mo id="S2.T1.3.1.1.1.m1.1.1.1" xref="S2.T1.3.1.1.1.m1.1.1.1.cmml">=</mo><mn id="S2.T1.3.1.1.1.m1.1.1.3" xref="S2.T1.3.1.1.1.m1.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.3.1.1.1.m1.1b"><apply id="S2.T1.3.1.1.1.m1.1.1.cmml" xref="S2.T1.3.1.1.1.m1.1.1"><eq id="S2.T1.3.1.1.1.m1.1.1.1.cmml" xref="S2.T1.3.1.1.1.m1.1.1.1"></eq><ci id="S2.T1.3.1.1.1.m1.1.1.2.cmml" xref="S2.T1.3.1.1.1.m1.1.1.2">𝑎</ci><cn id="S2.T1.3.1.1.1.m1.1.1.3.cmml" type="float" xref="S2.T1.3.1.1.1.m1.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.3.1.1.1.m1.1c">a=0.1</annotation><annotation encoding="application/x-llamapun" id="S2.T1.3.1.1.1.m1.1d">italic_a = 0.1</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.4.2.2.2"><math alttext="a=0.01" class="ltx_Math" display="inline" id="S2.T1.4.2.2.2.m1.1"><semantics id="S2.T1.4.2.2.2.m1.1a"><mrow id="S2.T1.4.2.2.2.m1.1.1" xref="S2.T1.4.2.2.2.m1.1.1.cmml"><mi id="S2.T1.4.2.2.2.m1.1.1.2" xref="S2.T1.4.2.2.2.m1.1.1.2.cmml">a</mi><mo id="S2.T1.4.2.2.2.m1.1.1.1" xref="S2.T1.4.2.2.2.m1.1.1.1.cmml">=</mo><mn id="S2.T1.4.2.2.2.m1.1.1.3" xref="S2.T1.4.2.2.2.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.4.2.2.2.m1.1b"><apply id="S2.T1.4.2.2.2.m1.1.1.cmml" xref="S2.T1.4.2.2.2.m1.1.1"><eq id="S2.T1.4.2.2.2.m1.1.1.1.cmml" xref="S2.T1.4.2.2.2.m1.1.1.1"></eq><ci id="S2.T1.4.2.2.2.m1.1.1.2.cmml" xref="S2.T1.4.2.2.2.m1.1.1.2">𝑎</ci><cn id="S2.T1.4.2.2.2.m1.1.1.3.cmml" type="float" xref="S2.T1.4.2.2.2.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.4.2.2.2.m1.1c">a=0.01</annotation><annotation encoding="application/x-llamapun" id="S2.T1.4.2.2.2.m1.1d">italic_a = 0.01</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S2.T1.5.3.3.3"><math alttext="a=0.002" class="ltx_Math" display="inline" id="S2.T1.5.3.3.3.m1.1"><semantics id="S2.T1.5.3.3.3.m1.1a"><mrow id="S2.T1.5.3.3.3.m1.1.1" xref="S2.T1.5.3.3.3.m1.1.1.cmml"><mi id="S2.T1.5.3.3.3.m1.1.1.2" xref="S2.T1.5.3.3.3.m1.1.1.2.cmml">a</mi><mo id="S2.T1.5.3.3.3.m1.1.1.1" xref="S2.T1.5.3.3.3.m1.1.1.1.cmml">=</mo><mn id="S2.T1.5.3.3.3.m1.1.1.3" xref="S2.T1.5.3.3.3.m1.1.1.3.cmml">0.002</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.5.3.3.3.m1.1b"><apply id="S2.T1.5.3.3.3.m1.1.1.cmml" xref="S2.T1.5.3.3.3.m1.1.1"><eq id="S2.T1.5.3.3.3.m1.1.1.1.cmml" xref="S2.T1.5.3.3.3.m1.1.1.1"></eq><ci id="S2.T1.5.3.3.3.m1.1.1.2.cmml" xref="S2.T1.5.3.3.3.m1.1.1.2">𝑎</ci><cn id="S2.T1.5.3.3.3.m1.1.1.3.cmml" type="float" xref="S2.T1.5.3.3.3.m1.1.1.3">0.002</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.5.3.3.3.m1.1c">a=0.002</annotation><annotation encoding="application/x-llamapun" id="S2.T1.5.3.3.3.m1.1d">italic_a = 0.002</annotation></semantics></math></th> </tr> <tr class="ltx_tr" id="S2.T1.8.6.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r" id="S2.T1.8.6.6.4">case</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.6.4.4.1"><math alttext="N=10" class="ltx_Math" display="inline" id="S2.T1.6.4.4.1.m1.1"><semantics id="S2.T1.6.4.4.1.m1.1a"><mrow id="S2.T1.6.4.4.1.m1.1.1" xref="S2.T1.6.4.4.1.m1.1.1.cmml"><mi id="S2.T1.6.4.4.1.m1.1.1.2" xref="S2.T1.6.4.4.1.m1.1.1.2.cmml">N</mi><mo id="S2.T1.6.4.4.1.m1.1.1.1" xref="S2.T1.6.4.4.1.m1.1.1.1.cmml">=</mo><mn id="S2.T1.6.4.4.1.m1.1.1.3" xref="S2.T1.6.4.4.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.6.4.4.1.m1.1b"><apply id="S2.T1.6.4.4.1.m1.1.1.cmml" xref="S2.T1.6.4.4.1.m1.1.1"><eq id="S2.T1.6.4.4.1.m1.1.1.1.cmml" xref="S2.T1.6.4.4.1.m1.1.1.1"></eq><ci id="S2.T1.6.4.4.1.m1.1.1.2.cmml" xref="S2.T1.6.4.4.1.m1.1.1.2">𝑁</ci><cn id="S2.T1.6.4.4.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.6.4.4.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.6.4.4.1.m1.1c">N=10</annotation><annotation encoding="application/x-llamapun" id="S2.T1.6.4.4.1.m1.1d">italic_N = 10</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.7.5.5.2"><math alttext="N=100" class="ltx_Math" display="inline" id="S2.T1.7.5.5.2.m1.1"><semantics id="S2.T1.7.5.5.2.m1.1a"><mrow id="S2.T1.7.5.5.2.m1.1.1" xref="S2.T1.7.5.5.2.m1.1.1.cmml"><mi id="S2.T1.7.5.5.2.m1.1.1.2" xref="S2.T1.7.5.5.2.m1.1.1.2.cmml">N</mi><mo id="S2.T1.7.5.5.2.m1.1.1.1" xref="S2.T1.7.5.5.2.m1.1.1.1.cmml">=</mo><mn id="S2.T1.7.5.5.2.m1.1.1.3" xref="S2.T1.7.5.5.2.m1.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.7.5.5.2.m1.1b"><apply id="S2.T1.7.5.5.2.m1.1.1.cmml" xref="S2.T1.7.5.5.2.m1.1.1"><eq id="S2.T1.7.5.5.2.m1.1.1.1.cmml" xref="S2.T1.7.5.5.2.m1.1.1.1"></eq><ci id="S2.T1.7.5.5.2.m1.1.1.2.cmml" xref="S2.T1.7.5.5.2.m1.1.1.2">𝑁</ci><cn id="S2.T1.7.5.5.2.m1.1.1.3.cmml" type="integer" xref="S2.T1.7.5.5.2.m1.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.7.5.5.2.m1.1c">N=100</annotation><annotation encoding="application/x-llamapun" id="S2.T1.7.5.5.2.m1.1d">italic_N = 100</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S2.T1.8.6.6.3"><math alttext="N=500" class="ltx_Math" display="inline" id="S2.T1.8.6.6.3.m1.1"><semantics id="S2.T1.8.6.6.3.m1.1a"><mrow id="S2.T1.8.6.6.3.m1.1.1" xref="S2.T1.8.6.6.3.m1.1.1.cmml"><mi id="S2.T1.8.6.6.3.m1.1.1.2" xref="S2.T1.8.6.6.3.m1.1.1.2.cmml">N</mi><mo id="S2.T1.8.6.6.3.m1.1.1.1" xref="S2.T1.8.6.6.3.m1.1.1.1.cmml">=</mo><mn id="S2.T1.8.6.6.3.m1.1.1.3" xref="S2.T1.8.6.6.3.m1.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.8.6.6.3.m1.1b"><apply id="S2.T1.8.6.6.3.m1.1.1.cmml" xref="S2.T1.8.6.6.3.m1.1.1"><eq id="S2.T1.8.6.6.3.m1.1.1.1.cmml" xref="S2.T1.8.6.6.3.m1.1.1.1"></eq><ci id="S2.T1.8.6.6.3.m1.1.1.2.cmml" xref="S2.T1.8.6.6.3.m1.1.1.2">𝑁</ci><cn id="S2.T1.8.6.6.3.m1.1.1.3.cmml" type="integer" xref="S2.T1.8.6.6.3.m1.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.8.6.6.3.m1.1c">N=500</annotation><annotation encoding="application/x-llamapun" id="S2.T1.8.6.6.3.m1.1d">italic_N = 500</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T1.9.7.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.9.7.7.1"><math alttext="\pi" class="ltx_Math" display="inline" id="S2.T1.9.7.7.1.m1.1"><semantics id="S2.T1.9.7.7.1.m1.1a"><mi id="S2.T1.9.7.7.1.m1.1.1" xref="S2.T1.9.7.7.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S2.T1.9.7.7.1.m1.1b"><ci id="S2.T1.9.7.7.1.m1.1.1.cmml" xref="S2.T1.9.7.7.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.9.7.7.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.9.7.7.1.m1.1d">italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.9.7.7.2">3.1286893</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.9.7.7.3">3.1414634</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T1.9.7.7.4">3.1415874</td> </tr> <tr class="ltx_tr" id="S2.T1.10.8.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.10.8.8.1"><math alttext="2\pi" class="ltx_Math" display="inline" id="S2.T1.10.8.8.1.m1.1"><semantics id="S2.T1.10.8.8.1.m1.1a"><mrow id="S2.T1.10.8.8.1.m1.1.1" xref="S2.T1.10.8.8.1.m1.1.1.cmml"><mn id="S2.T1.10.8.8.1.m1.1.1.2" xref="S2.T1.10.8.8.1.m1.1.1.2.cmml">2</mn><mo id="S2.T1.10.8.8.1.m1.1.1.1" xref="S2.T1.10.8.8.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.10.8.8.1.m1.1.1.3" xref="S2.T1.10.8.8.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.10.8.8.1.m1.1b"><apply id="S2.T1.10.8.8.1.m1.1.1.cmml" xref="S2.T1.10.8.8.1.m1.1.1"><times id="S2.T1.10.8.8.1.m1.1.1.1.cmml" xref="S2.T1.10.8.8.1.m1.1.1.1"></times><cn id="S2.T1.10.8.8.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.10.8.8.1.m1.1.1.2">2</cn><ci id="S2.T1.10.8.8.1.m1.1.1.3.cmml" xref="S2.T1.10.8.8.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.10.8.8.1.m1.1c">2\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.10.8.8.1.m1.1d">2 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.10.8.8.2">6.1803398</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.10.8.8.3">6.2821518</td> <td class="ltx_td ltx_align_center" id="S2.T1.10.8.8.4">6.2831439</td> </tr> <tr class="ltx_tr" id="S2.T1.11.9.9"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.11.9.9.1"><math alttext="3\pi" class="ltx_Math" display="inline" id="S2.T1.11.9.9.1.m1.1"><semantics id="S2.T1.11.9.9.1.m1.1a"><mrow id="S2.T1.11.9.9.1.m1.1.1" xref="S2.T1.11.9.9.1.m1.1.1.cmml"><mn id="S2.T1.11.9.9.1.m1.1.1.2" xref="S2.T1.11.9.9.1.m1.1.1.2.cmml">3</mn><mo id="S2.T1.11.9.9.1.m1.1.1.1" xref="S2.T1.11.9.9.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.11.9.9.1.m1.1.1.3" xref="S2.T1.11.9.9.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.11.9.9.1.m1.1b"><apply id="S2.T1.11.9.9.1.m1.1.1.cmml" xref="S2.T1.11.9.9.1.m1.1.1"><times id="S2.T1.11.9.9.1.m1.1.1.1.cmml" xref="S2.T1.11.9.9.1.m1.1.1.1"></times><cn id="S2.T1.11.9.9.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.11.9.9.1.m1.1.1.2">3</cn><ci id="S2.T1.11.9.9.1.m1.1.1.3.cmml" xref="S2.T1.11.9.9.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.11.9.9.1.m1.1c">3\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.11.9.9.1.m1.1d">3 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.11.9.9.2">9.0798099</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.11.9.9.3">9.4212901</td> <td class="ltx_td ltx_align_center" id="S2.T1.11.9.9.4">9.4246384</td> </tr> <tr class="ltx_tr" id="S2.T1.12.10.10"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.12.10.10.1"><math alttext="4\pi" class="ltx_Math" display="inline" id="S2.T1.12.10.10.1.m1.1"><semantics id="S2.T1.12.10.10.1.m1.1a"><mrow id="S2.T1.12.10.10.1.m1.1.1" xref="S2.T1.12.10.10.1.m1.1.1.cmml"><mn id="S2.T1.12.10.10.1.m1.1.1.2" xref="S2.T1.12.10.10.1.m1.1.1.2.cmml">4</mn><mo id="S2.T1.12.10.10.1.m1.1.1.1" xref="S2.T1.12.10.10.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.12.10.10.1.m1.1.1.3" xref="S2.T1.12.10.10.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.12.10.10.1.m1.1b"><apply id="S2.T1.12.10.10.1.m1.1.1.cmml" xref="S2.T1.12.10.10.1.m1.1.1"><times id="S2.T1.12.10.10.1.m1.1.1.1.cmml" xref="S2.T1.12.10.10.1.m1.1.1.1"></times><cn id="S2.T1.12.10.10.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.12.10.10.1.m1.1.1.2">4</cn><ci id="S2.T1.12.10.10.1.m1.1.1.3.cmml" xref="S2.T1.12.10.10.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.12.10.10.1.m1.1c">4\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.12.10.10.1.m1.1d">4 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.12.10.10.2">11.7557050</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.12.10.10.3">12.5581039</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.10.10.4">12.5660398</td> </tr> <tr class="ltx_tr" id="S2.T1.13.11.11"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.13.11.11.1"><math alttext="5\pi" class="ltx_Math" display="inline" id="S2.T1.13.11.11.1.m1.1"><semantics id="S2.T1.13.11.11.1.m1.1a"><mrow id="S2.T1.13.11.11.1.m1.1.1" xref="S2.T1.13.11.11.1.m1.1.1.cmml"><mn id="S2.T1.13.11.11.1.m1.1.1.2" xref="S2.T1.13.11.11.1.m1.1.1.2.cmml">5</mn><mo id="S2.T1.13.11.11.1.m1.1.1.1" xref="S2.T1.13.11.11.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.13.11.11.1.m1.1.1.3" xref="S2.T1.13.11.11.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.13.11.11.1.m1.1b"><apply id="S2.T1.13.11.11.1.m1.1.1.cmml" xref="S2.T1.13.11.11.1.m1.1.1"><times id="S2.T1.13.11.11.1.m1.1.1.1.cmml" xref="S2.T1.13.11.11.1.m1.1.1.1"></times><cn id="S2.T1.13.11.11.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.13.11.11.1.m1.1.1.2">5</cn><ci id="S2.T1.13.11.11.1.m1.1.1.3.cmml" xref="S2.T1.13.11.11.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.13.11.11.1.m1.1c">5\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.13.11.11.1.m1.1d">5 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.13.11.11.2">14.1421356</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.13.11.11.3">15.6918191</td> <td class="ltx_td ltx_align_center" id="S2.T1.13.11.11.4">15.7073173</td> </tr> </tbody> </table> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <table class="ltx_tabular ltx_figure_panel ltx_parbox ltx_align_middle" id="S2.T1.24.11" style="width:212.5pt;"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S2.T1.16.3.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r" id="S2.T1.16.3.3.4">Continuous</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.14.1.1.1"><math alttext="a=0.1" class="ltx_Math" display="inline" id="S2.T1.14.1.1.1.m1.1"><semantics id="S2.T1.14.1.1.1.m1.1a"><mrow id="S2.T1.14.1.1.1.m1.1.1" xref="S2.T1.14.1.1.1.m1.1.1.cmml"><mi id="S2.T1.14.1.1.1.m1.1.1.2" xref="S2.T1.14.1.1.1.m1.1.1.2.cmml">a</mi><mo id="S2.T1.14.1.1.1.m1.1.1.1" xref="S2.T1.14.1.1.1.m1.1.1.1.cmml">=</mo><mn id="S2.T1.14.1.1.1.m1.1.1.3" xref="S2.T1.14.1.1.1.m1.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.14.1.1.1.m1.1b"><apply id="S2.T1.14.1.1.1.m1.1.1.cmml" xref="S2.T1.14.1.1.1.m1.1.1"><eq id="S2.T1.14.1.1.1.m1.1.1.1.cmml" xref="S2.T1.14.1.1.1.m1.1.1.1"></eq><ci id="S2.T1.14.1.1.1.m1.1.1.2.cmml" xref="S2.T1.14.1.1.1.m1.1.1.2">𝑎</ci><cn id="S2.T1.14.1.1.1.m1.1.1.3.cmml" type="float" xref="S2.T1.14.1.1.1.m1.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.14.1.1.1.m1.1c">a=0.1</annotation><annotation encoding="application/x-llamapun" id="S2.T1.14.1.1.1.m1.1d">italic_a = 0.1</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.15.2.2.2"><math alttext="a=0.01" class="ltx_Math" display="inline" id="S2.T1.15.2.2.2.m1.1"><semantics id="S2.T1.15.2.2.2.m1.1a"><mrow id="S2.T1.15.2.2.2.m1.1.1" xref="S2.T1.15.2.2.2.m1.1.1.cmml"><mi id="S2.T1.15.2.2.2.m1.1.1.2" xref="S2.T1.15.2.2.2.m1.1.1.2.cmml">a</mi><mo id="S2.T1.15.2.2.2.m1.1.1.1" xref="S2.T1.15.2.2.2.m1.1.1.1.cmml">=</mo><mn id="S2.T1.15.2.2.2.m1.1.1.3" xref="S2.T1.15.2.2.2.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.15.2.2.2.m1.1b"><apply id="S2.T1.15.2.2.2.m1.1.1.cmml" xref="S2.T1.15.2.2.2.m1.1.1"><eq id="S2.T1.15.2.2.2.m1.1.1.1.cmml" xref="S2.T1.15.2.2.2.m1.1.1.1"></eq><ci id="S2.T1.15.2.2.2.m1.1.1.2.cmml" xref="S2.T1.15.2.2.2.m1.1.1.2">𝑎</ci><cn id="S2.T1.15.2.2.2.m1.1.1.3.cmml" type="float" xref="S2.T1.15.2.2.2.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.15.2.2.2.m1.1c">a=0.01</annotation><annotation encoding="application/x-llamapun" id="S2.T1.15.2.2.2.m1.1d">italic_a = 0.01</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S2.T1.16.3.3.3"><math alttext="a=0.002" class="ltx_Math" display="inline" id="S2.T1.16.3.3.3.m1.1"><semantics id="S2.T1.16.3.3.3.m1.1a"><mrow id="S2.T1.16.3.3.3.m1.1.1" xref="S2.T1.16.3.3.3.m1.1.1.cmml"><mi id="S2.T1.16.3.3.3.m1.1.1.2" xref="S2.T1.16.3.3.3.m1.1.1.2.cmml">a</mi><mo id="S2.T1.16.3.3.3.m1.1.1.1" xref="S2.T1.16.3.3.3.m1.1.1.1.cmml">=</mo><mn id="S2.T1.16.3.3.3.m1.1.1.3" xref="S2.T1.16.3.3.3.m1.1.1.3.cmml">0.002</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.16.3.3.3.m1.1b"><apply id="S2.T1.16.3.3.3.m1.1.1.cmml" xref="S2.T1.16.3.3.3.m1.1.1"><eq id="S2.T1.16.3.3.3.m1.1.1.1.cmml" xref="S2.T1.16.3.3.3.m1.1.1.1"></eq><ci id="S2.T1.16.3.3.3.m1.1.1.2.cmml" xref="S2.T1.16.3.3.3.m1.1.1.2">𝑎</ci><cn id="S2.T1.16.3.3.3.m1.1.1.3.cmml" type="float" xref="S2.T1.16.3.3.3.m1.1.1.3">0.002</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.16.3.3.3.m1.1c">a=0.002</annotation><annotation encoding="application/x-llamapun" id="S2.T1.16.3.3.3.m1.1d">italic_a = 0.002</annotation></semantics></math></th> </tr> <tr class="ltx_tr" id="S2.T1.19.6.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r" id="S2.T1.19.6.6.4">case</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.17.4.4.1"><math alttext="N=10" class="ltx_Math" display="inline" id="S2.T1.17.4.4.1.m1.1"><semantics id="S2.T1.17.4.4.1.m1.1a"><mrow id="S2.T1.17.4.4.1.m1.1.1" xref="S2.T1.17.4.4.1.m1.1.1.cmml"><mi id="S2.T1.17.4.4.1.m1.1.1.2" xref="S2.T1.17.4.4.1.m1.1.1.2.cmml">N</mi><mo id="S2.T1.17.4.4.1.m1.1.1.1" xref="S2.T1.17.4.4.1.m1.1.1.1.cmml">=</mo><mn id="S2.T1.17.4.4.1.m1.1.1.3" xref="S2.T1.17.4.4.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.17.4.4.1.m1.1b"><apply id="S2.T1.17.4.4.1.m1.1.1.cmml" xref="S2.T1.17.4.4.1.m1.1.1"><eq id="S2.T1.17.4.4.1.m1.1.1.1.cmml" xref="S2.T1.17.4.4.1.m1.1.1.1"></eq><ci id="S2.T1.17.4.4.1.m1.1.1.2.cmml" xref="S2.T1.17.4.4.1.m1.1.1.2">𝑁</ci><cn id="S2.T1.17.4.4.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.17.4.4.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.17.4.4.1.m1.1c">N=10</annotation><annotation encoding="application/x-llamapun" id="S2.T1.17.4.4.1.m1.1d">italic_N = 10</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S2.T1.18.5.5.2"><math alttext="N=100" class="ltx_Math" display="inline" id="S2.T1.18.5.5.2.m1.1"><semantics id="S2.T1.18.5.5.2.m1.1a"><mrow id="S2.T1.18.5.5.2.m1.1.1" xref="S2.T1.18.5.5.2.m1.1.1.cmml"><mi id="S2.T1.18.5.5.2.m1.1.1.2" xref="S2.T1.18.5.5.2.m1.1.1.2.cmml">N</mi><mo id="S2.T1.18.5.5.2.m1.1.1.1" xref="S2.T1.18.5.5.2.m1.1.1.1.cmml">=</mo><mn id="S2.T1.18.5.5.2.m1.1.1.3" xref="S2.T1.18.5.5.2.m1.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.18.5.5.2.m1.1b"><apply id="S2.T1.18.5.5.2.m1.1.1.cmml" xref="S2.T1.18.5.5.2.m1.1.1"><eq id="S2.T1.18.5.5.2.m1.1.1.1.cmml" xref="S2.T1.18.5.5.2.m1.1.1.1"></eq><ci id="S2.T1.18.5.5.2.m1.1.1.2.cmml" xref="S2.T1.18.5.5.2.m1.1.1.2">𝑁</ci><cn id="S2.T1.18.5.5.2.m1.1.1.3.cmml" type="integer" xref="S2.T1.18.5.5.2.m1.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.18.5.5.2.m1.1c">N=100</annotation><annotation encoding="application/x-llamapun" id="S2.T1.18.5.5.2.m1.1d">italic_N = 100</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S2.T1.19.6.6.3"><math alttext="N=500" class="ltx_Math" display="inline" id="S2.T1.19.6.6.3.m1.1"><semantics id="S2.T1.19.6.6.3.m1.1a"><mrow id="S2.T1.19.6.6.3.m1.1.1" xref="S2.T1.19.6.6.3.m1.1.1.cmml"><mi id="S2.T1.19.6.6.3.m1.1.1.2" xref="S2.T1.19.6.6.3.m1.1.1.2.cmml">N</mi><mo id="S2.T1.19.6.6.3.m1.1.1.1" xref="S2.T1.19.6.6.3.m1.1.1.1.cmml">=</mo><mn id="S2.T1.19.6.6.3.m1.1.1.3" xref="S2.T1.19.6.6.3.m1.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.19.6.6.3.m1.1b"><apply id="S2.T1.19.6.6.3.m1.1.1.cmml" xref="S2.T1.19.6.6.3.m1.1.1"><eq id="S2.T1.19.6.6.3.m1.1.1.1.cmml" xref="S2.T1.19.6.6.3.m1.1.1.1"></eq><ci id="S2.T1.19.6.6.3.m1.1.1.2.cmml" xref="S2.T1.19.6.6.3.m1.1.1.2">𝑁</ci><cn id="S2.T1.19.6.6.3.m1.1.1.3.cmml" type="integer" xref="S2.T1.19.6.6.3.m1.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.19.6.6.3.m1.1c">N=500</annotation><annotation encoding="application/x-llamapun" id="S2.T1.19.6.6.3.m1.1d">italic_N = 500</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T1.20.7.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.20.7.7.1"><math alttext="\pi" class="ltx_Math" display="inline" id="S2.T1.20.7.7.1.m1.1"><semantics id="S2.T1.20.7.7.1.m1.1a"><mi id="S2.T1.20.7.7.1.m1.1.1" xref="S2.T1.20.7.7.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S2.T1.20.7.7.1.m1.1b"><ci id="S2.T1.20.7.7.1.m1.1.1.cmml" xref="S2.T1.20.7.7.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.20.7.7.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.20.7.7.1.m1.1d">italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.20.7.7.2">3.4729635</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.20.7.7.3">3.1731927</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T1.20.7.7.4">3.1478832</td> </tr> <tr class="ltx_tr" id="S2.T1.21.8.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.21.8.8.1"><math alttext="2\pi" class="ltx_Math" display="inline" id="S2.T1.21.8.8.1.m1.1"><semantics id="S2.T1.21.8.8.1.m1.1a"><mrow id="S2.T1.21.8.8.1.m1.1.1" xref="S2.T1.21.8.8.1.m1.1.1.cmml"><mn id="S2.T1.21.8.8.1.m1.1.1.2" xref="S2.T1.21.8.8.1.m1.1.1.2.cmml">2</mn><mo id="S2.T1.21.8.8.1.m1.1.1.1" xref="S2.T1.21.8.8.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.21.8.8.1.m1.1.1.3" xref="S2.T1.21.8.8.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.21.8.8.1.m1.1b"><apply id="S2.T1.21.8.8.1.m1.1.1.cmml" xref="S2.T1.21.8.8.1.m1.1.1"><times id="S2.T1.21.8.8.1.m1.1.1.1.cmml" xref="S2.T1.21.8.8.1.m1.1.1.1"></times><cn id="S2.T1.21.8.8.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.21.8.8.1.m1.1.1.2">2</cn><ci id="S2.T1.21.8.8.1.m1.1.1.3.cmml" xref="S2.T1.21.8.8.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.21.8.8.1.m1.1c">2\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.21.8.8.1.m1.1d">2 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.21.8.8.2">6.8404028</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.21.8.8.3">6.3455866</td> <td class="ltx_td ltx_align_center" id="S2.T1.21.8.8.4">6.2957352</td> </tr> <tr class="ltx_tr" id="S2.T1.22.9.9"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.22.9.9.1"><math alttext="3\pi" class="ltx_Math" display="inline" id="S2.T1.22.9.9.1.m1.1"><semantics id="S2.T1.22.9.9.1.m1.1a"><mrow id="S2.T1.22.9.9.1.m1.1.1" xref="S2.T1.22.9.9.1.m1.1.1.cmml"><mn id="S2.T1.22.9.9.1.m1.1.1.2" xref="S2.T1.22.9.9.1.m1.1.1.2.cmml">3</mn><mo id="S2.T1.22.9.9.1.m1.1.1.1" xref="S2.T1.22.9.9.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.22.9.9.1.m1.1.1.3" xref="S2.T1.22.9.9.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.22.9.9.1.m1.1b"><apply id="S2.T1.22.9.9.1.m1.1.1.cmml" xref="S2.T1.22.9.9.1.m1.1.1"><times id="S2.T1.22.9.9.1.m1.1.1.1.cmml" xref="S2.T1.22.9.9.1.m1.1.1.1"></times><cn id="S2.T1.22.9.9.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.22.9.9.1.m1.1.1.2">3</cn><ci id="S2.T1.22.9.9.1.m1.1.1.3.cmml" xref="S2.T1.22.9.9.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.22.9.9.1.m1.1c">3\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.22.9.9.1.m1.1d">3 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.22.9.9.2">9.9999999</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.22.9.9.3">9.5163831</td> <td class="ltx_td ltx_align_center" id="S2.T1.22.9.9.4">9.4435249</td> </tr> <tr class="ltx_tr" id="S2.T1.23.10.10"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.23.10.10.1"><math alttext="4\pi" class="ltx_Math" display="inline" id="S2.T1.23.10.10.1.m1.1"><semantics id="S2.T1.23.10.10.1.m1.1a"><mrow id="S2.T1.23.10.10.1.m1.1.1" xref="S2.T1.23.10.10.1.m1.1.1.cmml"><mn id="S2.T1.23.10.10.1.m1.1.1.2" xref="S2.T1.23.10.10.1.m1.1.1.2.cmml">4</mn><mo id="S2.T1.23.10.10.1.m1.1.1.1" xref="S2.T1.23.10.10.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.23.10.10.1.m1.1.1.3" xref="S2.T1.23.10.10.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.23.10.10.1.m1.1b"><apply id="S2.T1.23.10.10.1.m1.1.1.cmml" xref="S2.T1.23.10.10.1.m1.1.1"><times id="S2.T1.23.10.10.1.m1.1.1.1.cmml" xref="S2.T1.23.10.10.1.m1.1.1.1"></times><cn id="S2.T1.23.10.10.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.23.10.10.1.m1.1.1.2">4</cn><ci id="S2.T1.23.10.10.1.m1.1.1.3.cmml" xref="S2.T1.23.10.10.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.23.10.10.1.m1.1c">4\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.23.10.10.1.m1.1d">4 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.23.10.10.2">12.8557521</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.23.10.10.3">12.6847839</td> <td class="ltx_td ltx_align_center" id="S2.T1.23.10.10.4">12.5912209</td> </tr> <tr class="ltx_tr" id="S2.T1.24.11.11"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S2.T1.24.11.11.1"><math alttext="5\pi" class="ltx_Math" display="inline" id="S2.T1.24.11.11.1.m1.1"><semantics id="S2.T1.24.11.11.1.m1.1a"><mrow id="S2.T1.24.11.11.1.m1.1.1" xref="S2.T1.24.11.11.1.m1.1.1.cmml"><mn id="S2.T1.24.11.11.1.m1.1.1.2" xref="S2.T1.24.11.11.1.m1.1.1.2.cmml">5</mn><mo id="S2.T1.24.11.11.1.m1.1.1.1" xref="S2.T1.24.11.11.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.T1.24.11.11.1.m1.1.1.3" xref="S2.T1.24.11.11.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.24.11.11.1.m1.1b"><apply id="S2.T1.24.11.11.1.m1.1.1.cmml" xref="S2.T1.24.11.11.1.m1.1.1"><times id="S2.T1.24.11.11.1.m1.1.1.1.cmml" xref="S2.T1.24.11.11.1.m1.1.1.1"></times><cn id="S2.T1.24.11.11.1.m1.1.1.2.cmml" type="integer" xref="S2.T1.24.11.11.1.m1.1.1.2">5</cn><ci id="S2.T1.24.11.11.1.m1.1.1.3.cmml" xref="S2.T1.24.11.11.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.24.11.11.1.m1.1c">5\pi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.24.11.11.1.m1.1d">5 italic_π</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.24.11.11.2">15.3208888</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.T1.24.11.11.3">15.8499913</td> <td class="ltx_td ltx_align_center" id="S2.T1.24.11.11.4">15.7387923</td> </tr> </tbody> </table> </div> </div> </figure> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.5">Now let’s consider the problem on the finite discrete interval <math alttext="[0,L]" class="ltx_Math" display="inline" id="S2.p4.1.m1.2"><semantics id="S2.p4.1.m1.2a"><mrow id="S2.p4.1.m1.2.3.2" xref="S2.p4.1.m1.2.3.1.cmml"><mo id="S2.p4.1.m1.2.3.2.1" stretchy="false" xref="S2.p4.1.m1.2.3.1.cmml">[</mo><mn id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml">0</mn><mo id="S2.p4.1.m1.2.3.2.2" xref="S2.p4.1.m1.2.3.1.cmml">,</mo><mi id="S2.p4.1.m1.2.2" xref="S2.p4.1.m1.2.2.cmml">L</mi><mo id="S2.p4.1.m1.2.3.2.3" stretchy="false" xref="S2.p4.1.m1.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.2b"><interval closure="closed" id="S2.p4.1.m1.2.3.1.cmml" xref="S2.p4.1.m1.2.3.2"><cn id="S2.p4.1.m1.1.1.cmml" type="integer" xref="S2.p4.1.m1.1.1">0</cn><ci id="S2.p4.1.m1.2.2.cmml" xref="S2.p4.1.m1.2.2">𝐿</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.2c">[0,L]</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.2d">[ 0 , italic_L ]</annotation></semantics></math> with step size <math alttext="a=L/N" class="ltx_Math" display="inline" id="S2.p4.2.m2.1"><semantics id="S2.p4.2.m2.1a"><mrow id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml"><mi id="S2.p4.2.m2.1.1.2" xref="S2.p4.2.m2.1.1.2.cmml">a</mi><mo id="S2.p4.2.m2.1.1.1" xref="S2.p4.2.m2.1.1.1.cmml">=</mo><mrow id="S2.p4.2.m2.1.1.3" xref="S2.p4.2.m2.1.1.3.cmml"><mi id="S2.p4.2.m2.1.1.3.2" xref="S2.p4.2.m2.1.1.3.2.cmml">L</mi><mo id="S2.p4.2.m2.1.1.3.1" xref="S2.p4.2.m2.1.1.3.1.cmml">/</mo><mi id="S2.p4.2.m2.1.1.3.3" xref="S2.p4.2.m2.1.1.3.3.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.2.m2.1b"><apply id="S2.p4.2.m2.1.1.cmml" xref="S2.p4.2.m2.1.1"><eq id="S2.p4.2.m2.1.1.1.cmml" xref="S2.p4.2.m2.1.1.1"></eq><ci id="S2.p4.2.m2.1.1.2.cmml" xref="S2.p4.2.m2.1.1.2">𝑎</ci><apply id="S2.p4.2.m2.1.1.3.cmml" xref="S2.p4.2.m2.1.1.3"><divide id="S2.p4.2.m2.1.1.3.1.cmml" xref="S2.p4.2.m2.1.1.3.1"></divide><ci id="S2.p4.2.m2.1.1.3.2.cmml" xref="S2.p4.2.m2.1.1.3.2">𝐿</ci><ci id="S2.p4.2.m2.1.1.3.3.cmml" xref="S2.p4.2.m2.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.2.m2.1c">a=L/N</annotation><annotation encoding="application/x-llamapun" id="S2.p4.2.m2.1d">italic_a = italic_L / italic_N</annotation></semantics></math> for some <math alttext="N\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.p4.3.m3.1"><semantics id="S2.p4.3.m3.1a"><mrow id="S2.p4.3.m3.1.1" xref="S2.p4.3.m3.1.1.cmml"><mi id="S2.p4.3.m3.1.1.2" xref="S2.p4.3.m3.1.1.2.cmml">N</mi><mo id="S2.p4.3.m3.1.1.1" xref="S2.p4.3.m3.1.1.1.cmml">∈</mo><mi id="S2.p4.3.m3.1.1.3" xref="S2.p4.3.m3.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.3.m3.1b"><apply id="S2.p4.3.m3.1.1.cmml" xref="S2.p4.3.m3.1.1"><in id="S2.p4.3.m3.1.1.1.cmml" xref="S2.p4.3.m3.1.1.1"></in><ci id="S2.p4.3.m3.1.1.2.cmml" xref="S2.p4.3.m3.1.1.2">𝑁</ci><ci id="S2.p4.3.m3.1.1.3.cmml" xref="S2.p4.3.m3.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.3.m3.1c">N\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.3.m3.1d">italic_N ∈ blackboard_N</annotation></semantics></math> and <math alttext="x_{n}^{(a)}=na" class="ltx_Math" display="inline" id="S2.p4.4.m4.1"><semantics id="S2.p4.4.m4.1a"><mrow id="S2.p4.4.m4.1.2" xref="S2.p4.4.m4.1.2.cmml"><msubsup id="S2.p4.4.m4.1.2.2" xref="S2.p4.4.m4.1.2.2.cmml"><mi id="S2.p4.4.m4.1.2.2.2.2" xref="S2.p4.4.m4.1.2.2.2.2.cmml">x</mi><mi id="S2.p4.4.m4.1.2.2.2.3" xref="S2.p4.4.m4.1.2.2.2.3.cmml">n</mi><mrow id="S2.p4.4.m4.1.1.1.3" xref="S2.p4.4.m4.1.2.2.cmml"><mo id="S2.p4.4.m4.1.1.1.3.1" stretchy="false" xref="S2.p4.4.m4.1.2.2.cmml">(</mo><mi id="S2.p4.4.m4.1.1.1.1" xref="S2.p4.4.m4.1.1.1.1.cmml">a</mi><mo id="S2.p4.4.m4.1.1.1.3.2" stretchy="false" xref="S2.p4.4.m4.1.2.2.cmml">)</mo></mrow></msubsup><mo id="S2.p4.4.m4.1.2.1" xref="S2.p4.4.m4.1.2.1.cmml">=</mo><mrow id="S2.p4.4.m4.1.2.3" xref="S2.p4.4.m4.1.2.3.cmml"><mi id="S2.p4.4.m4.1.2.3.2" xref="S2.p4.4.m4.1.2.3.2.cmml">n</mi><mo id="S2.p4.4.m4.1.2.3.1" xref="S2.p4.4.m4.1.2.3.1.cmml">⁢</mo><mi id="S2.p4.4.m4.1.2.3.3" xref="S2.p4.4.m4.1.2.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.4.m4.1b"><apply id="S2.p4.4.m4.1.2.cmml" xref="S2.p4.4.m4.1.2"><eq id="S2.p4.4.m4.1.2.1.cmml" xref="S2.p4.4.m4.1.2.1"></eq><apply id="S2.p4.4.m4.1.2.2.cmml" xref="S2.p4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.p4.4.m4.1.2.2.1.cmml" xref="S2.p4.4.m4.1.2.2">superscript</csymbol><apply id="S2.p4.4.m4.1.2.2.2.cmml" xref="S2.p4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.p4.4.m4.1.2.2.2.1.cmml" xref="S2.p4.4.m4.1.2.2">subscript</csymbol><ci id="S2.p4.4.m4.1.2.2.2.2.cmml" xref="S2.p4.4.m4.1.2.2.2.2">𝑥</ci><ci id="S2.p4.4.m4.1.2.2.2.3.cmml" xref="S2.p4.4.m4.1.2.2.2.3">𝑛</ci></apply><ci id="S2.p4.4.m4.1.1.1.1.cmml" xref="S2.p4.4.m4.1.1.1.1">𝑎</ci></apply><apply id="S2.p4.4.m4.1.2.3.cmml" xref="S2.p4.4.m4.1.2.3"><times id="S2.p4.4.m4.1.2.3.1.cmml" xref="S2.p4.4.m4.1.2.3.1"></times><ci id="S2.p4.4.m4.1.2.3.2.cmml" xref="S2.p4.4.m4.1.2.3.2">𝑛</ci><ci id="S2.p4.4.m4.1.2.3.3.cmml" xref="S2.p4.4.m4.1.2.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.4.m4.1c">x_{n}^{(a)}=na</annotation><annotation encoding="application/x-llamapun" id="S2.p4.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT = italic_n italic_a</annotation></semantics></math> for <math alttext="n=0,...,N" class="ltx_Math" display="inline" id="S2.p4.5.m5.3"><semantics id="S2.p4.5.m5.3a"><mrow id="S2.p4.5.m5.3.4" xref="S2.p4.5.m5.3.4.cmml"><mi id="S2.p4.5.m5.3.4.2" xref="S2.p4.5.m5.3.4.2.cmml">n</mi><mo id="S2.p4.5.m5.3.4.1" xref="S2.p4.5.m5.3.4.1.cmml">=</mo><mrow id="S2.p4.5.m5.3.4.3.2" xref="S2.p4.5.m5.3.4.3.1.cmml"><mn id="S2.p4.5.m5.1.1" xref="S2.p4.5.m5.1.1.cmml">0</mn><mo id="S2.p4.5.m5.3.4.3.2.1" xref="S2.p4.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.p4.5.m5.2.2" mathvariant="normal" xref="S2.p4.5.m5.2.2.cmml">…</mi><mo id="S2.p4.5.m5.3.4.3.2.2" xref="S2.p4.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.p4.5.m5.3.3" xref="S2.p4.5.m5.3.3.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.5.m5.3b"><apply id="S2.p4.5.m5.3.4.cmml" xref="S2.p4.5.m5.3.4"><eq id="S2.p4.5.m5.3.4.1.cmml" xref="S2.p4.5.m5.3.4.1"></eq><ci id="S2.p4.5.m5.3.4.2.cmml" xref="S2.p4.5.m5.3.4.2">𝑛</ci><list id="S2.p4.5.m5.3.4.3.1.cmml" xref="S2.p4.5.m5.3.4.3.2"><cn id="S2.p4.5.m5.1.1.cmml" type="integer" xref="S2.p4.5.m5.1.1">0</cn><ci id="S2.p4.5.m5.2.2.cmml" xref="S2.p4.5.m5.2.2">…</ci><ci id="S2.p4.5.m5.3.3.cmml" xref="S2.p4.5.m5.3.3">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.5.m5.3c">n=0,...,N</annotation><annotation encoding="application/x-llamapun" id="S2.p4.5.m5.3d">italic_n = 0 , … , italic_N</annotation></semantics></math>. Imposing Dirichlet boundary condition as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx4"> <tbody id="S2.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi^{(a)}(0)=0,\quad\Psi^{(a)}(Na)=0" class="ltx_Math" display="inline" id="S2.E6.m1.5"><semantics id="S2.E6.m1.5a"><mrow id="S2.E6.m1.5.5.2" xref="S2.E6.m1.5.5.3.cmml"><mrow id="S2.E6.m1.4.4.1.1" xref="S2.E6.m1.4.4.1.1.cmml"><mrow id="S2.E6.m1.4.4.1.1.2" xref="S2.E6.m1.4.4.1.1.2.cmml"><msup id="S2.E6.m1.4.4.1.1.2.2" xref="S2.E6.m1.4.4.1.1.2.2.cmml"><mi id="S2.E6.m1.4.4.1.1.2.2.2" mathvariant="normal" xref="S2.E6.m1.4.4.1.1.2.2.2.cmml">Ψ</mi><mrow id="S2.E6.m1.1.1.1.3" xref="S2.E6.m1.4.4.1.1.2.2.cmml"><mo id="S2.E6.m1.1.1.1.3.1" stretchy="false" xref="S2.E6.m1.4.4.1.1.2.2.cmml">(</mo><mi id="S2.E6.m1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.cmml">a</mi><mo id="S2.E6.m1.1.1.1.3.2" stretchy="false" xref="S2.E6.m1.4.4.1.1.2.2.cmml">)</mo></mrow></msup><mo id="S2.E6.m1.4.4.1.1.2.1" xref="S2.E6.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S2.E6.m1.4.4.1.1.2.3.2" xref="S2.E6.m1.4.4.1.1.2.cmml"><mo id="S2.E6.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S2.E6.m1.4.4.1.1.2.cmml">(</mo><mn id="S2.E6.m1.3.3" xref="S2.E6.m1.3.3.cmml">0</mn><mo id="S2.E6.m1.4.4.1.1.2.3.2.2" stretchy="false" xref="S2.E6.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.4.4.1.1.1" xref="S2.E6.m1.4.4.1.1.1.cmml">=</mo><mn id="S2.E6.m1.4.4.1.1.3" xref="S2.E6.m1.4.4.1.1.3.cmml">0</mn></mrow><mo id="S2.E6.m1.5.5.2.3" rspace="1.167em" xref="S2.E6.m1.5.5.3a.cmml">,</mo><mrow id="S2.E6.m1.5.5.2.2" xref="S2.E6.m1.5.5.2.2.cmml"><mrow id="S2.E6.m1.5.5.2.2.1" xref="S2.E6.m1.5.5.2.2.1.cmml"><msup id="S2.E6.m1.5.5.2.2.1.3" xref="S2.E6.m1.5.5.2.2.1.3.cmml"><mi id="S2.E6.m1.5.5.2.2.1.3.2" mathvariant="normal" xref="S2.E6.m1.5.5.2.2.1.3.2.cmml">Ψ</mi><mrow id="S2.E6.m1.2.2.1.3" xref="S2.E6.m1.5.5.2.2.1.3.cmml"><mo id="S2.E6.m1.2.2.1.3.1" stretchy="false" xref="S2.E6.m1.5.5.2.2.1.3.cmml">(</mo><mi id="S2.E6.m1.2.2.1.1" xref="S2.E6.m1.2.2.1.1.cmml">a</mi><mo id="S2.E6.m1.2.2.1.3.2" stretchy="false" xref="S2.E6.m1.5.5.2.2.1.3.cmml">)</mo></mrow></msup><mo id="S2.E6.m1.5.5.2.2.1.2" xref="S2.E6.m1.5.5.2.2.1.2.cmml">⁢</mo><mrow id="S2.E6.m1.5.5.2.2.1.1.1" xref="S2.E6.m1.5.5.2.2.1.1.1.1.cmml"><mo id="S2.E6.m1.5.5.2.2.1.1.1.2" stretchy="false" xref="S2.E6.m1.5.5.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.E6.m1.5.5.2.2.1.1.1.1" xref="S2.E6.m1.5.5.2.2.1.1.1.1.cmml"><mi id="S2.E6.m1.5.5.2.2.1.1.1.1.2" xref="S2.E6.m1.5.5.2.2.1.1.1.1.2.cmml">N</mi><mo id="S2.E6.m1.5.5.2.2.1.1.1.1.1" xref="S2.E6.m1.5.5.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E6.m1.5.5.2.2.1.1.1.1.3" xref="S2.E6.m1.5.5.2.2.1.1.1.1.3.cmml">a</mi></mrow><mo id="S2.E6.m1.5.5.2.2.1.1.1.3" stretchy="false" xref="S2.E6.m1.5.5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.5.5.2.2.2" xref="S2.E6.m1.5.5.2.2.2.cmml">=</mo><mn id="S2.E6.m1.5.5.2.2.3" xref="S2.E6.m1.5.5.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E6.m1.5b"><apply id="S2.E6.m1.5.5.3.cmml" xref="S2.E6.m1.5.5.2"><csymbol cd="ambiguous" id="S2.E6.m1.5.5.3a.cmml" xref="S2.E6.m1.5.5.2.3">formulae-sequence</csymbol><apply id="S2.E6.m1.4.4.1.1.cmml" xref="S2.E6.m1.4.4.1.1"><eq id="S2.E6.m1.4.4.1.1.1.cmml" xref="S2.E6.m1.4.4.1.1.1"></eq><apply id="S2.E6.m1.4.4.1.1.2.cmml" xref="S2.E6.m1.4.4.1.1.2"><times id="S2.E6.m1.4.4.1.1.2.1.cmml" xref="S2.E6.m1.4.4.1.1.2.1"></times><apply id="S2.E6.m1.4.4.1.1.2.2.cmml" xref="S2.E6.m1.4.4.1.1.2.2"><csymbol cd="ambiguous" id="S2.E6.m1.4.4.1.1.2.2.1.cmml" xref="S2.E6.m1.4.4.1.1.2.2">superscript</csymbol><ci id="S2.E6.m1.4.4.1.1.2.2.2.cmml" xref="S2.E6.m1.4.4.1.1.2.2.2">Ψ</ci><ci id="S2.E6.m1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1">𝑎</ci></apply><cn id="S2.E6.m1.3.3.cmml" type="integer" xref="S2.E6.m1.3.3">0</cn></apply><cn id="S2.E6.m1.4.4.1.1.3.cmml" type="integer" xref="S2.E6.m1.4.4.1.1.3">0</cn></apply><apply id="S2.E6.m1.5.5.2.2.cmml" xref="S2.E6.m1.5.5.2.2"><eq id="S2.E6.m1.5.5.2.2.2.cmml" xref="S2.E6.m1.5.5.2.2.2"></eq><apply id="S2.E6.m1.5.5.2.2.1.cmml" xref="S2.E6.m1.5.5.2.2.1"><times id="S2.E6.m1.5.5.2.2.1.2.cmml" xref="S2.E6.m1.5.5.2.2.1.2"></times><apply id="S2.E6.m1.5.5.2.2.1.3.cmml" xref="S2.E6.m1.5.5.2.2.1.3"><csymbol cd="ambiguous" id="S2.E6.m1.5.5.2.2.1.3.1.cmml" xref="S2.E6.m1.5.5.2.2.1.3">superscript</csymbol><ci id="S2.E6.m1.5.5.2.2.1.3.2.cmml" xref="S2.E6.m1.5.5.2.2.1.3.2">Ψ</ci><ci id="S2.E6.m1.2.2.1.1.cmml" xref="S2.E6.m1.2.2.1.1">𝑎</ci></apply><apply id="S2.E6.m1.5.5.2.2.1.1.1.1.cmml" xref="S2.E6.m1.5.5.2.2.1.1.1"><times id="S2.E6.m1.5.5.2.2.1.1.1.1.1.cmml" xref="S2.E6.m1.5.5.2.2.1.1.1.1.1"></times><ci id="S2.E6.m1.5.5.2.2.1.1.1.1.2.cmml" xref="S2.E6.m1.5.5.2.2.1.1.1.1.2">𝑁</ci><ci id="S2.E6.m1.5.5.2.2.1.1.1.1.3.cmml" xref="S2.E6.m1.5.5.2.2.1.1.1.1.3">𝑎</ci></apply></apply><cn id="S2.E6.m1.5.5.2.2.3.cmml" type="integer" xref="S2.E6.m1.5.5.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.5c">\displaystyle\Psi^{(a)}(0)=0,\quad\Psi^{(a)}(Na)=0</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.5d">roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( 0 ) = 0 , roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_N italic_a ) = 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p4.7">leads to the following homogeneous system of linear equations for <math alttext="A" class="ltx_Math" display="inline" id="S2.p4.6.m1.1"><semantics id="S2.p4.6.m1.1a"><mi id="S2.p4.6.m1.1.1" xref="S2.p4.6.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.p4.6.m1.1b"><ci id="S2.p4.6.m1.1.1.cmml" xref="S2.p4.6.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.6.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.p4.6.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S2.p4.7.m2.1"><semantics id="S2.p4.7.m2.1a"><mi id="S2.p4.7.m2.1.1" xref="S2.p4.7.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.p4.7.m2.1b"><ci id="S2.p4.7.m2.1.1.cmml" xref="S2.p4.7.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.7.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.p4.7.m2.1d">italic_B</annotation></semantics></math>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx5"> <tbody id="S2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}&A+B=0,\\ &Ag_{+}(a)^{N}+Bg_{-}(a)^{N}=0.\end{split}" class="ltx_Math" display="inline" id="S2.E7.m1.27"><semantics id="S2.E7.m1.27a"><mtable columnspacing="0pt" id="S2.E7.m1.27.27.3" rowspacing="0pt"><mtr id="S2.E7.m1.27.27.3a"><mtd id="S2.E7.m1.27.27.3b"></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.E7.m1.27.27.3c"><mrow id="S2.E7.m1.26.26.2.25.7.7.7"><mrow id="S2.E7.m1.26.26.2.25.7.7.7.1"><mrow id="S2.E7.m1.26.26.2.25.7.7.7.1.1"><mi id="S2.E7.m1.1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.cmml">A</mi><mo id="S2.E7.m1.2.2.2.2.2.2" xref="S2.E7.m1.2.2.2.2.2.2.cmml">+</mo><mi id="S2.E7.m1.3.3.3.3.3.3" xref="S2.E7.m1.3.3.3.3.3.3.cmml">B</mi></mrow><mo id="S2.E7.m1.4.4.4.4.4.4" xref="S2.E7.m1.4.4.4.4.4.4.cmml">=</mo><mn id="S2.E7.m1.5.5.5.5.5.5" xref="S2.E7.m1.5.5.5.5.5.5.cmml">0</mn></mrow><mo id="S2.E7.m1.6.6.6.6.6.6">,</mo></mrow></mtd></mtr><mtr id="S2.E7.m1.27.27.3d"><mtd id="S2.E7.m1.27.27.3e"></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.E7.m1.27.27.3f"><mrow id="S2.E7.m1.27.27.3.26.19.19.19"><mrow id="S2.E7.m1.27.27.3.26.19.19.19.1"><mrow id="S2.E7.m1.27.27.3.26.19.19.19.1.1"><mrow id="S2.E7.m1.27.27.3.26.19.19.19.1.1.1"><mi id="S2.E7.m1.7.7.7.1.1.1" xref="S2.E7.m1.7.7.7.1.1.1.cmml">A</mi><mo id="S2.E7.m1.27.27.3.26.19.19.19.1.1.1.1">⁢</mo><msub id="S2.E7.m1.27.27.3.26.19.19.19.1.1.1.2"><mi id="S2.E7.m1.8.8.8.2.2.2" xref="S2.E7.m1.8.8.8.2.2.2.cmml">g</mi><mo id="S2.E7.m1.9.9.9.3.3.3.1" xref="S2.E7.m1.9.9.9.3.3.3.1.cmml">+</mo></msub><mo id="S2.E7.m1.27.27.3.26.19.19.19.1.1.1.1a">⁢</mo><msup id="S2.E7.m1.27.27.3.26.19.19.19.1.1.1.3"><mrow id="S2.E7.m1.27.27.3.26.19.19.19.1.1.1.3.2"><mo id="S2.E7.m1.10.10.10.4.4.4" stretchy="false">(</mo><mi id="S2.E7.m1.11.11.11.5.5.5" xref="S2.E7.m1.11.11.11.5.5.5.cmml">a</mi><mo id="S2.E7.m1.12.12.12.6.6.6" stretchy="false">)</mo></mrow><mi id="S2.E7.m1.13.13.13.7.7.7.1" 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&Ag_{+}(a)^{N}+Bg_{-}(a)^{N}=0.\end{split}</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.27d">start_ROW start_CELL end_CELL start_CELL italic_A + italic_B = 0 , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_A italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT + italic_B italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT = 0 . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p4.8">The existence of non-trivial solutions of the above algebraic system leads to the following secular equation with respect to <math alttext="k" class="ltx_Math" display="inline" id="S2.p4.8.m1.1"><semantics id="S2.p4.8.m1.1a"><mi id="S2.p4.8.m1.1.1" xref="S2.p4.8.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p4.8.m1.1b"><ci id="S2.p4.8.m1.1.1.cmml" xref="S2.p4.8.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.8.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p4.8.m1.1d">italic_k</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="S2.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{+}(a)^{N}=g_{-}(a)^{N}." class="ltx_Math" display="block" id="S2.E8.m1.3"><semantics id="S2.E8.m1.3a"><mrow id="S2.E8.m1.3.3.1" xref="S2.E8.m1.3.3.1.1.cmml"><mrow id="S2.E8.m1.3.3.1.1" xref="S2.E8.m1.3.3.1.1.cmml"><mrow id="S2.E8.m1.3.3.1.1.2" xref="S2.E8.m1.3.3.1.1.2.cmml"><msub id="S2.E8.m1.3.3.1.1.2.2" xref="S2.E8.m1.3.3.1.1.2.2.cmml"><mi 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xref="S2.E8.m1.3.3.1.1.3.2.2">𝑔</ci><minus id="S2.E8.m1.3.3.1.1.3.2.3.cmml" xref="S2.E8.m1.3.3.1.1.3.2.3"></minus></apply><apply id="S2.E8.m1.3.3.1.1.3.3.cmml" xref="S2.E8.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="S2.E8.m1.3.3.1.1.3.3.1.cmml" xref="S2.E8.m1.3.3.1.1.3.3">superscript</csymbol><ci id="S2.E8.m1.2.2.cmml" xref="S2.E8.m1.2.2">𝑎</ci><ci id="S2.E8.m1.3.3.1.1.3.3.3.cmml" xref="S2.E8.m1.3.3.1.1.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.3c">g_{+}(a)^{N}=g_{-}(a)^{N}.</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.3d">italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT = italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.2">First, we consider the case <math alttext="k^{2}a^{2}>4" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><mrow id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml"><mrow id="S2.p5.1.m1.1.1.2" xref="S2.p5.1.m1.1.1.2.cmml"><msup id="S2.p5.1.m1.1.1.2.2" xref="S2.p5.1.m1.1.1.2.2.cmml"><mi id="S2.p5.1.m1.1.1.2.2.2" xref="S2.p5.1.m1.1.1.2.2.2.cmml">k</mi><mn id="S2.p5.1.m1.1.1.2.2.3" xref="S2.p5.1.m1.1.1.2.2.3.cmml">2</mn></msup><mo id="S2.p5.1.m1.1.1.2.1" xref="S2.p5.1.m1.1.1.2.1.cmml">⁢</mo><msup id="S2.p5.1.m1.1.1.2.3" xref="S2.p5.1.m1.1.1.2.3.cmml"><mi id="S2.p5.1.m1.1.1.2.3.2" xref="S2.p5.1.m1.1.1.2.3.2.cmml">a</mi><mn id="S2.p5.1.m1.1.1.2.3.3" xref="S2.p5.1.m1.1.1.2.3.3.cmml">2</mn></msup></mrow><mo id="S2.p5.1.m1.1.1.1" xref="S2.p5.1.m1.1.1.1.cmml">></mo><mn id="S2.p5.1.m1.1.1.3" xref="S2.p5.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><apply id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1"><gt id="S2.p5.1.m1.1.1.1.cmml" xref="S2.p5.1.m1.1.1.1"></gt><apply id="S2.p5.1.m1.1.1.2.cmml" xref="S2.p5.1.m1.1.1.2"><times id="S2.p5.1.m1.1.1.2.1.cmml" xref="S2.p5.1.m1.1.1.2.1"></times><apply id="S2.p5.1.m1.1.1.2.2.cmml" xref="S2.p5.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.2.2.1.cmml" xref="S2.p5.1.m1.1.1.2.2">superscript</csymbol><ci id="S2.p5.1.m1.1.1.2.2.2.cmml" xref="S2.p5.1.m1.1.1.2.2.2">𝑘</ci><cn id="S2.p5.1.m1.1.1.2.2.3.cmml" type="integer" xref="S2.p5.1.m1.1.1.2.2.3">2</cn></apply><apply id="S2.p5.1.m1.1.1.2.3.cmml" xref="S2.p5.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.2.3.1.cmml" xref="S2.p5.1.m1.1.1.2.3">superscript</csymbol><ci id="S2.p5.1.m1.1.1.2.3.2.cmml" xref="S2.p5.1.m1.1.1.2.3.2">𝑎</ci><cn id="S2.p5.1.m1.1.1.2.3.3.cmml" type="integer" xref="S2.p5.1.m1.1.1.2.3.3">2</cn></apply></apply><cn id="S2.p5.1.m1.1.1.3.cmml" type="integer" xref="S2.p5.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">k^{2}a^{2}>4</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT > 4</annotation></semantics></math> and <math alttext="k>0" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><mrow id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml"><mi id="S2.p5.2.m2.1.1.2" xref="S2.p5.2.m2.1.1.2.cmml">k</mi><mo id="S2.p5.2.m2.1.1.1" xref="S2.p5.2.m2.1.1.1.cmml">></mo><mn id="S2.p5.2.m2.1.1.3" xref="S2.p5.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><apply id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1"><gt id="S2.p5.2.m2.1.1.1.cmml" xref="S2.p5.2.m2.1.1.1"></gt><ci id="S2.p5.2.m2.1.1.2.cmml" xref="S2.p5.2.m2.1.1.2">𝑘</ci><cn id="S2.p5.2.m2.1.1.3.cmml" type="integer" xref="S2.p5.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">k>0</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">italic_k > 0</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{+}(a)-g_{-}(a)=2ka\sqrt{k^{2}a^{2}-4}>0" class="ltx_Math" display="block" id="S2.Ex3.m1.2"><semantics id="S2.Ex3.m1.2a"><mrow id="S2.Ex3.m1.2.3" xref="S2.Ex3.m1.2.3.cmml"><mrow id="S2.Ex3.m1.2.3.2" xref="S2.Ex3.m1.2.3.2.cmml"><mrow id="S2.Ex3.m1.2.3.2.2" xref="S2.Ex3.m1.2.3.2.2.cmml"><msub id="S2.Ex3.m1.2.3.2.2.2" xref="S2.Ex3.m1.2.3.2.2.2.cmml"><mi id="S2.Ex3.m1.2.3.2.2.2.2" xref="S2.Ex3.m1.2.3.2.2.2.2.cmml">g</mi><mo id="S2.Ex3.m1.2.3.2.2.2.3" xref="S2.Ex3.m1.2.3.2.2.2.3.cmml">+</mo></msub><mo id="S2.Ex3.m1.2.3.2.2.1" xref="S2.Ex3.m1.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.3.2.2.3.2" xref="S2.Ex3.m1.2.3.2.2.cmml"><mo id="S2.Ex3.m1.2.3.2.2.3.2.1" stretchy="false" xref="S2.Ex3.m1.2.3.2.2.cmml">(</mo><mi id="S2.Ex3.m1.1.1" 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id="S2.Ex3.m1.2d">italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) - italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) = 2 italic_k italic_a square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 4 end_ARG > 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p5.5">and <math alttext="g_{+}(a)>g_{-}(a)" class="ltx_Math" display="inline" id="S2.p5.3.m1.2"><semantics id="S2.p5.3.m1.2a"><mrow id="S2.p5.3.m1.2.3" xref="S2.p5.3.m1.2.3.cmml"><mrow id="S2.p5.3.m1.2.3.2" xref="S2.p5.3.m1.2.3.2.cmml"><msub id="S2.p5.3.m1.2.3.2.2" xref="S2.p5.3.m1.2.3.2.2.cmml"><mi id="S2.p5.3.m1.2.3.2.2.2" xref="S2.p5.3.m1.2.3.2.2.2.cmml">g</mi><mo id="S2.p5.3.m1.2.3.2.2.3" xref="S2.p5.3.m1.2.3.2.2.3.cmml">+</mo></msub><mo id="S2.p5.3.m1.2.3.2.1" xref="S2.p5.3.m1.2.3.2.1.cmml">⁢</mo><mrow id="S2.p5.3.m1.2.3.2.3.2" xref="S2.p5.3.m1.2.3.2.cmml"><mo id="S2.p5.3.m1.2.3.2.3.2.1" stretchy="false" xref="S2.p5.3.m1.2.3.2.cmml">(</mo><mi id="S2.p5.3.m1.1.1" xref="S2.p5.3.m1.1.1.cmml">a</mi><mo id="S2.p5.3.m1.2.3.2.3.2.2" stretchy="false" xref="S2.p5.3.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.p5.3.m1.2.3.1" xref="S2.p5.3.m1.2.3.1.cmml">></mo><mrow id="S2.p5.3.m1.2.3.3" xref="S2.p5.3.m1.2.3.3.cmml"><msub id="S2.p5.3.m1.2.3.3.2" xref="S2.p5.3.m1.2.3.3.2.cmml"><mi id="S2.p5.3.m1.2.3.3.2.2" xref="S2.p5.3.m1.2.3.3.2.2.cmml">g</mi><mo id="S2.p5.3.m1.2.3.3.2.3" xref="S2.p5.3.m1.2.3.3.2.3.cmml">−</mo></msub><mo id="S2.p5.3.m1.2.3.3.1" xref="S2.p5.3.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.p5.3.m1.2.3.3.3.2" xref="S2.p5.3.m1.2.3.3.cmml"><mo id="S2.p5.3.m1.2.3.3.3.2.1" stretchy="false" xref="S2.p5.3.m1.2.3.3.cmml">(</mo><mi id="S2.p5.3.m1.2.2" xref="S2.p5.3.m1.2.2.cmml">a</mi><mo id="S2.p5.3.m1.2.3.3.3.2.2" stretchy="false" xref="S2.p5.3.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.3.m1.2b"><apply id="S2.p5.3.m1.2.3.cmml" xref="S2.p5.3.m1.2.3"><gt id="S2.p5.3.m1.2.3.1.cmml" xref="S2.p5.3.m1.2.3.1"></gt><apply id="S2.p5.3.m1.2.3.2.cmml" xref="S2.p5.3.m1.2.3.2"><times id="S2.p5.3.m1.2.3.2.1.cmml" xref="S2.p5.3.m1.2.3.2.1"></times><apply id="S2.p5.3.m1.2.3.2.2.cmml" xref="S2.p5.3.m1.2.3.2.2"><csymbol cd="ambiguous" id="S2.p5.3.m1.2.3.2.2.1.cmml" xref="S2.p5.3.m1.2.3.2.2">subscript</csymbol><ci id="S2.p5.3.m1.2.3.2.2.2.cmml" xref="S2.p5.3.m1.2.3.2.2.2">𝑔</ci><plus id="S2.p5.3.m1.2.3.2.2.3.cmml" xref="S2.p5.3.m1.2.3.2.2.3"></plus></apply><ci id="S2.p5.3.m1.1.1.cmml" xref="S2.p5.3.m1.1.1">𝑎</ci></apply><apply id="S2.p5.3.m1.2.3.3.cmml" xref="S2.p5.3.m1.2.3.3"><times id="S2.p5.3.m1.2.3.3.1.cmml" xref="S2.p5.3.m1.2.3.3.1"></times><apply id="S2.p5.3.m1.2.3.3.2.cmml" xref="S2.p5.3.m1.2.3.3.2"><csymbol cd="ambiguous" id="S2.p5.3.m1.2.3.3.2.1.cmml" xref="S2.p5.3.m1.2.3.3.2">subscript</csymbol><ci id="S2.p5.3.m1.2.3.3.2.2.cmml" xref="S2.p5.3.m1.2.3.3.2.2">𝑔</ci><minus id="S2.p5.3.m1.2.3.3.2.3.cmml" xref="S2.p5.3.m1.2.3.3.2.3"></minus></apply><ci id="S2.p5.3.m1.2.2.cmml" xref="S2.p5.3.m1.2.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.3.m1.2c">g_{+}(a)>g_{-}(a)</annotation><annotation encoding="application/x-llamapun" id="S2.p5.3.m1.2d">italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) > italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math> follows. Hence (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E8" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">8</span></a>) does not hold. A similar arguments shows the same for the case <math alttext="k^{2}a^{2}>4" class="ltx_Math" display="inline" id="S2.p5.4.m2.1"><semantics id="S2.p5.4.m2.1a"><mrow id="S2.p5.4.m2.1.1" xref="S2.p5.4.m2.1.1.cmml"><mrow id="S2.p5.4.m2.1.1.2" xref="S2.p5.4.m2.1.1.2.cmml"><msup id="S2.p5.4.m2.1.1.2.2" xref="S2.p5.4.m2.1.1.2.2.cmml"><mi id="S2.p5.4.m2.1.1.2.2.2" xref="S2.p5.4.m2.1.1.2.2.2.cmml">k</mi><mn id="S2.p5.4.m2.1.1.2.2.3" xref="S2.p5.4.m2.1.1.2.2.3.cmml">2</mn></msup><mo id="S2.p5.4.m2.1.1.2.1" xref="S2.p5.4.m2.1.1.2.1.cmml">⁢</mo><msup id="S2.p5.4.m2.1.1.2.3" xref="S2.p5.4.m2.1.1.2.3.cmml"><mi id="S2.p5.4.m2.1.1.2.3.2" xref="S2.p5.4.m2.1.1.2.3.2.cmml">a</mi><mn id="S2.p5.4.m2.1.1.2.3.3" xref="S2.p5.4.m2.1.1.2.3.3.cmml">2</mn></msup></mrow><mo id="S2.p5.4.m2.1.1.1" xref="S2.p5.4.m2.1.1.1.cmml">></mo><mn id="S2.p5.4.m2.1.1.3" xref="S2.p5.4.m2.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.4.m2.1b"><apply id="S2.p5.4.m2.1.1.cmml" xref="S2.p5.4.m2.1.1"><gt id="S2.p5.4.m2.1.1.1.cmml" xref="S2.p5.4.m2.1.1.1"></gt><apply id="S2.p5.4.m2.1.1.2.cmml" xref="S2.p5.4.m2.1.1.2"><times id="S2.p5.4.m2.1.1.2.1.cmml" xref="S2.p5.4.m2.1.1.2.1"></times><apply id="S2.p5.4.m2.1.1.2.2.cmml" xref="S2.p5.4.m2.1.1.2.2"><csymbol cd="ambiguous" id="S2.p5.4.m2.1.1.2.2.1.cmml" xref="S2.p5.4.m2.1.1.2.2">superscript</csymbol><ci id="S2.p5.4.m2.1.1.2.2.2.cmml" xref="S2.p5.4.m2.1.1.2.2.2">𝑘</ci><cn id="S2.p5.4.m2.1.1.2.2.3.cmml" type="integer" xref="S2.p5.4.m2.1.1.2.2.3">2</cn></apply><apply id="S2.p5.4.m2.1.1.2.3.cmml" xref="S2.p5.4.m2.1.1.2.3"><csymbol cd="ambiguous" id="S2.p5.4.m2.1.1.2.3.1.cmml" xref="S2.p5.4.m2.1.1.2.3">superscript</csymbol><ci id="S2.p5.4.m2.1.1.2.3.2.cmml" xref="S2.p5.4.m2.1.1.2.3.2">𝑎</ci><cn id="S2.p5.4.m2.1.1.2.3.3.cmml" type="integer" xref="S2.p5.4.m2.1.1.2.3.3">2</cn></apply></apply><cn id="S2.p5.4.m2.1.1.3.cmml" type="integer" xref="S2.p5.4.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.4.m2.1c">k^{2}a^{2}>4</annotation><annotation encoding="application/x-llamapun" id="S2.p5.4.m2.1d">italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT > 4</annotation></semantics></math> and <math alttext="k<0" class="ltx_Math" display="inline" id="S2.p5.5.m3.1"><semantics id="S2.p5.5.m3.1a"><mrow id="S2.p5.5.m3.1.1" xref="S2.p5.5.m3.1.1.cmml"><mi id="S2.p5.5.m3.1.1.2" xref="S2.p5.5.m3.1.1.2.cmml">k</mi><mo id="S2.p5.5.m3.1.1.1" xref="S2.p5.5.m3.1.1.1.cmml"><</mo><mn id="S2.p5.5.m3.1.1.3" xref="S2.p5.5.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.5.m3.1b"><apply id="S2.p5.5.m3.1.1.cmml" xref="S2.p5.5.m3.1.1"><lt id="S2.p5.5.m3.1.1.1.cmml" xref="S2.p5.5.m3.1.1.1"></lt><ci id="S2.p5.5.m3.1.1.2.cmml" xref="S2.p5.5.m3.1.1.2">𝑘</ci><cn id="S2.p5.5.m3.1.1.3.cmml" type="integer" xref="S2.p5.5.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.5.m3.1c">k<0</annotation><annotation encoding="application/x-llamapun" id="S2.p5.5.m3.1d">italic_k < 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">It remains to consider the case <math alttext="k^{2}a^{2}\leq 4" class="ltx_Math" display="inline" id="S2.p6.1.m1.1"><semantics id="S2.p6.1.m1.1a"><mrow id="S2.p6.1.m1.1.1" xref="S2.p6.1.m1.1.1.cmml"><mrow id="S2.p6.1.m1.1.1.2" xref="S2.p6.1.m1.1.1.2.cmml"><msup id="S2.p6.1.m1.1.1.2.2" xref="S2.p6.1.m1.1.1.2.2.cmml"><mi id="S2.p6.1.m1.1.1.2.2.2" xref="S2.p6.1.m1.1.1.2.2.2.cmml">k</mi><mn id="S2.p6.1.m1.1.1.2.2.3" xref="S2.p6.1.m1.1.1.2.2.3.cmml">2</mn></msup><mo id="S2.p6.1.m1.1.1.2.1" xref="S2.p6.1.m1.1.1.2.1.cmml">⁢</mo><msup id="S2.p6.1.m1.1.1.2.3" xref="S2.p6.1.m1.1.1.2.3.cmml"><mi id="S2.p6.1.m1.1.1.2.3.2" xref="S2.p6.1.m1.1.1.2.3.2.cmml">a</mi><mn id="S2.p6.1.m1.1.1.2.3.3" xref="S2.p6.1.m1.1.1.2.3.3.cmml">2</mn></msup></mrow><mo id="S2.p6.1.m1.1.1.1" xref="S2.p6.1.m1.1.1.1.cmml">≤</mo><mn id="S2.p6.1.m1.1.1.3" xref="S2.p6.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.1.m1.1b"><apply id="S2.p6.1.m1.1.1.cmml" xref="S2.p6.1.m1.1.1"><leq id="S2.p6.1.m1.1.1.1.cmml" xref="S2.p6.1.m1.1.1.1"></leq><apply id="S2.p6.1.m1.1.1.2.cmml" xref="S2.p6.1.m1.1.1.2"><times id="S2.p6.1.m1.1.1.2.1.cmml" xref="S2.p6.1.m1.1.1.2.1"></times><apply id="S2.p6.1.m1.1.1.2.2.cmml" xref="S2.p6.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S2.p6.1.m1.1.1.2.2.1.cmml" xref="S2.p6.1.m1.1.1.2.2">superscript</csymbol><ci id="S2.p6.1.m1.1.1.2.2.2.cmml" xref="S2.p6.1.m1.1.1.2.2.2">𝑘</ci><cn id="S2.p6.1.m1.1.1.2.2.3.cmml" type="integer" xref="S2.p6.1.m1.1.1.2.2.3">2</cn></apply><apply id="S2.p6.1.m1.1.1.2.3.cmml" xref="S2.p6.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S2.p6.1.m1.1.1.2.3.1.cmml" xref="S2.p6.1.m1.1.1.2.3">superscript</csymbol><ci id="S2.p6.1.m1.1.1.2.3.2.cmml" xref="S2.p6.1.m1.1.1.2.3.2">𝑎</ci><cn id="S2.p6.1.m1.1.1.2.3.3.cmml" type="integer" xref="S2.p6.1.m1.1.1.2.3.3">2</cn></apply></apply><cn id="S2.p6.1.m1.1.1.3.cmml" type="integer" xref="S2.p6.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.1.m1.1c">k^{2}a^{2}\leq 4</annotation><annotation encoding="application/x-llamapun" id="S2.p6.1.m1.1d">italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 4</annotation></semantics></math>. Then</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx6"> <tbody id="S2.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle g_{\pm}(a)=1+\frac{-k^{2}a^{2}\pm ika\sqrt{4-k^{2}a^{2}}}{2}." class="ltx_Math" display="inline" id="S2.E9.m1.2"><semantics id="S2.E9.m1.2a"><mrow id="S2.E9.m1.2.2.1" xref="S2.E9.m1.2.2.1.1.cmml"><mrow id="S2.E9.m1.2.2.1.1" xref="S2.E9.m1.2.2.1.1.cmml"><mrow id="S2.E9.m1.2.2.1.1.2" xref="S2.E9.m1.2.2.1.1.2.cmml"><msub id="S2.E9.m1.2.2.1.1.2.2" xref="S2.E9.m1.2.2.1.1.2.2.cmml"><mi id="S2.E9.m1.2.2.1.1.2.2.2" xref="S2.E9.m1.2.2.1.1.2.2.2.cmml">g</mi><mo id="S2.E9.m1.2.2.1.1.2.2.3" xref="S2.E9.m1.2.2.1.1.2.2.3.cmml">±</mo></msub><mo id="S2.E9.m1.2.2.1.1.2.1" xref="S2.E9.m1.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S2.E9.m1.2.2.1.1.2.3.2" xref="S2.E9.m1.2.2.1.1.2.cmml"><mo id="S2.E9.m1.2.2.1.1.2.3.2.1" stretchy="false" xref="S2.E9.m1.2.2.1.1.2.cmml">(</mo><mi id="S2.E9.m1.1.1" xref="S2.E9.m1.1.1.cmml">a</mi><mo id="S2.E9.m1.2.2.1.1.2.3.2.2" stretchy="false" xref="S2.E9.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E9.m1.2.2.1.1.1" xref="S2.E9.m1.2.2.1.1.1.cmml">=</mo><mrow id="S2.E9.m1.2.2.1.1.3" xref="S2.E9.m1.2.2.1.1.3.cmml"><mn id="S2.E9.m1.2.2.1.1.3.2" xref="S2.E9.m1.2.2.1.1.3.2.cmml">1</mn><mo id="S2.E9.m1.2.2.1.1.3.1" xref="S2.E9.m1.2.2.1.1.3.1.cmml">+</mo><mstyle displaystyle="true" id="S2.E9.m1.2.2.1.1.3.3" xref="S2.E9.m1.2.2.1.1.3.3.cmml"><mfrac id="S2.E9.m1.2.2.1.1.3.3a" xref="S2.E9.m1.2.2.1.1.3.3.cmml"><mrow id="S2.E9.m1.2.2.1.1.3.3.2" xref="S2.E9.m1.2.2.1.1.3.3.2.cmml"><mrow id="S2.E9.m1.2.2.1.1.3.3.2.2" xref="S2.E9.m1.2.2.1.1.3.3.2.2.cmml"><mo id="S2.E9.m1.2.2.1.1.3.3.2.2a" xref="S2.E9.m1.2.2.1.1.3.3.2.2.cmml">−</mo><mrow id="S2.E9.m1.2.2.1.1.3.3.2.2.2" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.cmml"><msup id="S2.E9.m1.2.2.1.1.3.3.2.2.2.2" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.cmml"><mi id="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.2" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.2.cmml">k</mi><mn id="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.3" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.3.cmml">2</mn></msup><mo id="S2.E9.m1.2.2.1.1.3.3.2.2.2.1" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.1.cmml">⁢</mo><msup id="S2.E9.m1.2.2.1.1.3.3.2.2.2.3" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.cmml"><mi id="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.2" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.2.cmml">a</mi><mn id="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.3" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S2.E9.m1.2.2.1.1.3.3.2.1" xref="S2.E9.m1.2.2.1.1.3.3.2.1.cmml">±</mo><mrow id="S2.E9.m1.2.2.1.1.3.3.2.3" xref="S2.E9.m1.2.2.1.1.3.3.2.3.cmml"><mi id="S2.E9.m1.2.2.1.1.3.3.2.3.2" xref="S2.E9.m1.2.2.1.1.3.3.2.3.2.cmml">i</mi><mo id="S2.E9.m1.2.2.1.1.3.3.2.3.1" xref="S2.E9.m1.2.2.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="S2.E9.m1.2.2.1.1.3.3.2.3.3" xref="S2.E9.m1.2.2.1.1.3.3.2.3.3.cmml">k</mi><mo id="S2.E9.m1.2.2.1.1.3.3.2.3.1a" xref="S2.E9.m1.2.2.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="S2.E9.m1.2.2.1.1.3.3.2.3.4" xref="S2.E9.m1.2.2.1.1.3.3.2.3.4.cmml">a</mi><mo id="S2.E9.m1.2.2.1.1.3.3.2.3.1b" xref="S2.E9.m1.2.2.1.1.3.3.2.3.1.cmml">⁢</mo><msqrt id="S2.E9.m1.2.2.1.1.3.3.2.3.5" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.cmml"><mrow id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.cmml"><mn id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.2" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.2.cmml">4</mn><mo id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.1" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.1.cmml">−</mo><mrow id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.cmml"><msup id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.cmml"><mi id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.2" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.2.cmml">k</mi><mn id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.3" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.3.cmml">2</mn></msup><mo id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.1" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.1.cmml">⁢</mo><msup id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.cmml"><mi id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.2" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.2.cmml">a</mi><mn id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.3" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.3.cmml">2</mn></msup></mrow></mrow></msqrt></mrow></mrow><mn id="S2.E9.m1.2.2.1.1.3.3.3" xref="S2.E9.m1.2.2.1.1.3.3.3.cmml">2</mn></mfrac></mstyle></mrow></mrow><mo id="S2.E9.m1.2.2.1.2" lspace="0em" xref="S2.E9.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E9.m1.2b"><apply id="S2.E9.m1.2.2.1.1.cmml" xref="S2.E9.m1.2.2.1"><eq id="S2.E9.m1.2.2.1.1.1.cmml" xref="S2.E9.m1.2.2.1.1.1"></eq><apply id="S2.E9.m1.2.2.1.1.2.cmml" xref="S2.E9.m1.2.2.1.1.2"><times id="S2.E9.m1.2.2.1.1.2.1.cmml" xref="S2.E9.m1.2.2.1.1.2.1"></times><apply id="S2.E9.m1.2.2.1.1.2.2.cmml" xref="S2.E9.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S2.E9.m1.2.2.1.1.2.2.1.cmml" xref="S2.E9.m1.2.2.1.1.2.2">subscript</csymbol><ci id="S2.E9.m1.2.2.1.1.2.2.2.cmml" xref="S2.E9.m1.2.2.1.1.2.2.2">𝑔</ci><csymbol cd="latexml" id="S2.E9.m1.2.2.1.1.2.2.3.cmml" xref="S2.E9.m1.2.2.1.1.2.2.3">plus-or-minus</csymbol></apply><ci id="S2.E9.m1.1.1.cmml" xref="S2.E9.m1.1.1">𝑎</ci></apply><apply id="S2.E9.m1.2.2.1.1.3.cmml" xref="S2.E9.m1.2.2.1.1.3"><plus id="S2.E9.m1.2.2.1.1.3.1.cmml" xref="S2.E9.m1.2.2.1.1.3.1"></plus><cn id="S2.E9.m1.2.2.1.1.3.2.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.3.2">1</cn><apply id="S2.E9.m1.2.2.1.1.3.3.cmml" xref="S2.E9.m1.2.2.1.1.3.3"><divide id="S2.E9.m1.2.2.1.1.3.3.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3"></divide><apply id="S2.E9.m1.2.2.1.1.3.3.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2"><csymbol cd="latexml" id="S2.E9.m1.2.2.1.1.3.3.2.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.1">plus-or-minus</csymbol><apply id="S2.E9.m1.2.2.1.1.3.3.2.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2"><minus id="S2.E9.m1.2.2.1.1.3.3.2.2.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2"></minus><apply id="S2.E9.m1.2.2.1.1.3.3.2.2.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2"><times id="S2.E9.m1.2.2.1.1.3.3.2.2.2.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.1"></times><apply id="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.2">superscript</csymbol><ci id="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.2">𝑘</ci><cn id="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.3.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.2.3">2</cn></apply><apply id="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.3">superscript</csymbol><ci id="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.2">𝑎</ci><cn id="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.3.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.3.3.2.2.2.3.3">2</cn></apply></apply></apply><apply id="S2.E9.m1.2.2.1.1.3.3.2.3.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3"><times id="S2.E9.m1.2.2.1.1.3.3.2.3.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.1"></times><ci id="S2.E9.m1.2.2.1.1.3.3.2.3.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.2">𝑖</ci><ci id="S2.E9.m1.2.2.1.1.3.3.2.3.3.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.3">𝑘</ci><ci id="S2.E9.m1.2.2.1.1.3.3.2.3.4.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.4">𝑎</ci><apply id="S2.E9.m1.2.2.1.1.3.3.2.3.5.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5"><root id="S2.E9.m1.2.2.1.1.3.3.2.3.5a.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5"></root><apply id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2"><minus id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.1"></minus><cn id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.2.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.2">4</cn><apply id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3"><times id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.1"></times><apply id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2"><csymbol cd="ambiguous" id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2">superscript</csymbol><ci id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.2">𝑘</ci><cn id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.3.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.2.3">2</cn></apply><apply id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3"><csymbol cd="ambiguous" id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.1.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3">superscript</csymbol><ci id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.2.cmml" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.2">𝑎</ci><cn id="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.3.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.3.3.2.3.5.2.3.3.3">2</cn></apply></apply></apply></apply></apply></apply><cn id="S2.E9.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.2c">\displaystyle g_{\pm}(a)=1+\frac{-k^{2}a^{2}\pm ika\sqrt{4-k^{2}a^{2}}}{2}.</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.2d">italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) = 1 + divide start_ARG - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ± italic_i italic_k italic_a square-root start_ARG 4 - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG start_ARG 2 end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.2">We write <math alttext="g_{\pm}(a)" class="ltx_Math" display="inline" id="S2.p6.2.m1.1"><semantics id="S2.p6.2.m1.1a"><mrow id="S2.p6.2.m1.1.2" xref="S2.p6.2.m1.1.2.cmml"><msub id="S2.p6.2.m1.1.2.2" xref="S2.p6.2.m1.1.2.2.cmml"><mi id="S2.p6.2.m1.1.2.2.2" xref="S2.p6.2.m1.1.2.2.2.cmml">g</mi><mo id="S2.p6.2.m1.1.2.2.3" xref="S2.p6.2.m1.1.2.2.3.cmml">±</mo></msub><mo id="S2.p6.2.m1.1.2.1" xref="S2.p6.2.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p6.2.m1.1.2.3.2" xref="S2.p6.2.m1.1.2.cmml"><mo id="S2.p6.2.m1.1.2.3.2.1" stretchy="false" xref="S2.p6.2.m1.1.2.cmml">(</mo><mi id="S2.p6.2.m1.1.1" xref="S2.p6.2.m1.1.1.cmml">a</mi><mo id="S2.p6.2.m1.1.2.3.2.2" stretchy="false" xref="S2.p6.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.2.m1.1b"><apply id="S2.p6.2.m1.1.2.cmml" xref="S2.p6.2.m1.1.2"><times id="S2.p6.2.m1.1.2.1.cmml" xref="S2.p6.2.m1.1.2.1"></times><apply id="S2.p6.2.m1.1.2.2.cmml" xref="S2.p6.2.m1.1.2.2"><csymbol cd="ambiguous" id="S2.p6.2.m1.1.2.2.1.cmml" xref="S2.p6.2.m1.1.2.2">subscript</csymbol><ci id="S2.p6.2.m1.1.2.2.2.cmml" xref="S2.p6.2.m1.1.2.2.2">𝑔</ci><csymbol cd="latexml" id="S2.p6.2.m1.1.2.2.3.cmml" xref="S2.p6.2.m1.1.2.2.3">plus-or-minus</csymbol></apply><ci id="S2.p6.2.m1.1.1.cmml" xref="S2.p6.2.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.2.m1.1c">g_{\pm}(a)</annotation><annotation encoding="application/x-llamapun" id="S2.p6.2.m1.1d">italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx7"> <tbody id="S2.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle g_{\pm}(a)=|g_{\pm}(a)|e^{i\varphi_{\pm}(a)}," class="ltx_Math" display="inline" id="S2.E10.m1.4"><semantics id="S2.E10.m1.4a"><mrow id="S2.E10.m1.4.4.1" xref="S2.E10.m1.4.4.1.1.cmml"><mrow id="S2.E10.m1.4.4.1.1" xref="S2.E10.m1.4.4.1.1.cmml"><mrow id="S2.E10.m1.4.4.1.1.3" xref="S2.E10.m1.4.4.1.1.3.cmml"><msub id="S2.E10.m1.4.4.1.1.3.2" xref="S2.E10.m1.4.4.1.1.3.2.cmml"><mi id="S2.E10.m1.4.4.1.1.3.2.2" xref="S2.E10.m1.4.4.1.1.3.2.2.cmml">g</mi><mo id="S2.E10.m1.4.4.1.1.3.2.3" xref="S2.E10.m1.4.4.1.1.3.2.3.cmml">±</mo></msub><mo id="S2.E10.m1.4.4.1.1.3.1" xref="S2.E10.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S2.E10.m1.4.4.1.1.3.3.2" xref="S2.E10.m1.4.4.1.1.3.cmml"><mo id="S2.E10.m1.4.4.1.1.3.3.2.1" stretchy="false" xref="S2.E10.m1.4.4.1.1.3.cmml">(</mo><mi id="S2.E10.m1.2.2" xref="S2.E10.m1.2.2.cmml">a</mi><mo id="S2.E10.m1.4.4.1.1.3.3.2.2" stretchy="false" xref="S2.E10.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E10.m1.4.4.1.1.2" xref="S2.E10.m1.4.4.1.1.2.cmml">=</mo><mrow id="S2.E10.m1.4.4.1.1.1" xref="S2.E10.m1.4.4.1.1.1.cmml"><mrow id="S2.E10.m1.4.4.1.1.1.1.1" xref="S2.E10.m1.4.4.1.1.1.1.2.cmml"><mo id="S2.E10.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S2.E10.m1.4.4.1.1.1.1.2.1.cmml">|</mo><mrow id="S2.E10.m1.4.4.1.1.1.1.1.1" xref="S2.E10.m1.4.4.1.1.1.1.1.1.cmml"><msub id="S2.E10.m1.4.4.1.1.1.1.1.1.2" xref="S2.E10.m1.4.4.1.1.1.1.1.1.2.cmml"><mi id="S2.E10.m1.4.4.1.1.1.1.1.1.2.2" xref="S2.E10.m1.4.4.1.1.1.1.1.1.2.2.cmml">g</mi><mo id="S2.E10.m1.4.4.1.1.1.1.1.1.2.3" xref="S2.E10.m1.4.4.1.1.1.1.1.1.2.3.cmml">±</mo></msub><mo id="S2.E10.m1.4.4.1.1.1.1.1.1.1" xref="S2.E10.m1.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.E10.m1.4.4.1.1.1.1.1.1.3.2" xref="S2.E10.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S2.E10.m1.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.E10.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="S2.E10.m1.3.3" xref="S2.E10.m1.3.3.cmml">a</mi><mo id="S2.E10.m1.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.E10.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E10.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="S2.E10.m1.4.4.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S2.E10.m1.4.4.1.1.1.2" xref="S2.E10.m1.4.4.1.1.1.2.cmml">⁢</mo><msup id="S2.E10.m1.4.4.1.1.1.3" xref="S2.E10.m1.4.4.1.1.1.3.cmml"><mi id="S2.E10.m1.4.4.1.1.1.3.2" xref="S2.E10.m1.4.4.1.1.1.3.2.cmml">e</mi><mrow id="S2.E10.m1.1.1.1" xref="S2.E10.m1.1.1.1.cmml"><mi id="S2.E10.m1.1.1.1.3" xref="S2.E10.m1.1.1.1.3.cmml">i</mi><mo id="S2.E10.m1.1.1.1.2" xref="S2.E10.m1.1.1.1.2.cmml">⁢</mo><msub id="S2.E10.m1.1.1.1.4" xref="S2.E10.m1.1.1.1.4.cmml"><mi id="S2.E10.m1.1.1.1.4.2" xref="S2.E10.m1.1.1.1.4.2.cmml">φ</mi><mo id="S2.E10.m1.1.1.1.4.3" xref="S2.E10.m1.1.1.1.4.3.cmml">±</mo></msub><mo id="S2.E10.m1.1.1.1.2a" xref="S2.E10.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E10.m1.1.1.1.5.2" xref="S2.E10.m1.1.1.1.cmml"><mo id="S2.E10.m1.1.1.1.5.2.1" stretchy="false" xref="S2.E10.m1.1.1.1.cmml">(</mo><mi id="S2.E10.m1.1.1.1.1" xref="S2.E10.m1.1.1.1.1.cmml">a</mi><mo id="S2.E10.m1.1.1.1.5.2.2" stretchy="false" xref="S2.E10.m1.1.1.1.cmml">)</mo></mrow></mrow></msup></mrow></mrow><mo id="S2.E10.m1.4.4.1.2" xref="S2.E10.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E10.m1.4b"><apply id="S2.E10.m1.4.4.1.1.cmml" xref="S2.E10.m1.4.4.1"><eq id="S2.E10.m1.4.4.1.1.2.cmml" xref="S2.E10.m1.4.4.1.1.2"></eq><apply id="S2.E10.m1.4.4.1.1.3.cmml" xref="S2.E10.m1.4.4.1.1.3"><times id="S2.E10.m1.4.4.1.1.3.1.cmml" xref="S2.E10.m1.4.4.1.1.3.1"></times><apply id="S2.E10.m1.4.4.1.1.3.2.cmml" xref="S2.E10.m1.4.4.1.1.3.2"><csymbol cd="ambiguous" id="S2.E10.m1.4.4.1.1.3.2.1.cmml" xref="S2.E10.m1.4.4.1.1.3.2">subscript</csymbol><ci id="S2.E10.m1.4.4.1.1.3.2.2.cmml" xref="S2.E10.m1.4.4.1.1.3.2.2">𝑔</ci><csymbol cd="latexml" id="S2.E10.m1.4.4.1.1.3.2.3.cmml" xref="S2.E10.m1.4.4.1.1.3.2.3">plus-or-minus</csymbol></apply><ci id="S2.E10.m1.2.2.cmml" xref="S2.E10.m1.2.2">𝑎</ci></apply><apply id="S2.E10.m1.4.4.1.1.1.cmml" xref="S2.E10.m1.4.4.1.1.1"><times id="S2.E10.m1.4.4.1.1.1.2.cmml" xref="S2.E10.m1.4.4.1.1.1.2"></times><apply id="S2.E10.m1.4.4.1.1.1.1.2.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1"><abs id="S2.E10.m1.4.4.1.1.1.1.2.1.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1.2"></abs><apply id="S2.E10.m1.4.4.1.1.1.1.1.1.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1.1"><times id="S2.E10.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1.1.1"></times><apply id="S2.E10.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E10.m1.4.4.1.1.1.1.1.1.2.1.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.E10.m1.4.4.1.1.1.1.1.1.2.2.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1.1.2.2">𝑔</ci><csymbol cd="latexml" id="S2.E10.m1.4.4.1.1.1.1.1.1.2.3.cmml" xref="S2.E10.m1.4.4.1.1.1.1.1.1.2.3">plus-or-minus</csymbol></apply><ci id="S2.E10.m1.3.3.cmml" xref="S2.E10.m1.3.3">𝑎</ci></apply></apply><apply id="S2.E10.m1.4.4.1.1.1.3.cmml" xref="S2.E10.m1.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S2.E10.m1.4.4.1.1.1.3.1.cmml" xref="S2.E10.m1.4.4.1.1.1.3">superscript</csymbol><ci id="S2.E10.m1.4.4.1.1.1.3.2.cmml" xref="S2.E10.m1.4.4.1.1.1.3.2">𝑒</ci><apply id="S2.E10.m1.1.1.1.cmml" xref="S2.E10.m1.1.1.1"><times id="S2.E10.m1.1.1.1.2.cmml" xref="S2.E10.m1.1.1.1.2"></times><ci id="S2.E10.m1.1.1.1.3.cmml" xref="S2.E10.m1.1.1.1.3">𝑖</ci><apply id="S2.E10.m1.1.1.1.4.cmml" xref="S2.E10.m1.1.1.1.4"><csymbol cd="ambiguous" id="S2.E10.m1.1.1.1.4.1.cmml" xref="S2.E10.m1.1.1.1.4">subscript</csymbol><ci id="S2.E10.m1.1.1.1.4.2.cmml" xref="S2.E10.m1.1.1.1.4.2">𝜑</ci><csymbol cd="latexml" id="S2.E10.m1.1.1.1.4.3.cmml" xref="S2.E10.m1.1.1.1.4.3">plus-or-minus</csymbol></apply><ci id="S2.E10.m1.1.1.1.1.cmml" xref="S2.E10.m1.1.1.1.1">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.4c">\displaystyle g_{\pm}(a)=|g_{\pm}(a)|e^{i\varphi_{\pm}(a)},</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.4d">italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) = | italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) | italic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.15">and we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx8"> <tbody id="S2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\text{Re}(g_{\pm}(a))=1-\frac{k^{2}a^{2}}{2}," class="ltx_Math" display="inline" id="S2.Ex4.m1.2"><semantics id="S2.Ex4.m1.2a"><mrow id="S2.Ex4.m1.2.2.1" xref="S2.Ex4.m1.2.2.1.1.cmml"><mrow id="S2.Ex4.m1.2.2.1.1" xref="S2.Ex4.m1.2.2.1.1.cmml"><mrow id="S2.Ex4.m1.2.2.1.1.1" xref="S2.Ex4.m1.2.2.1.1.1.cmml"><mtext id="S2.Ex4.m1.2.2.1.1.1.3" xref="S2.Ex4.m1.2.2.1.1.1.3a.cmml">Re</mtext><mo id="S2.Ex4.m1.2.2.1.1.1.2" xref="S2.Ex4.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.2.2.1.1.1.1.1" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex4.m1.2.2.1.1.1.1.1.1" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.cmml"><msub id="S2.Ex4.m1.2.2.1.1.1.1.1.1.2" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex4.m1.2.2.1.1.1.1.1.1.2.2" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.2.2.cmml">g</mi><mo id="S2.Ex4.m1.2.2.1.1.1.1.1.1.2.3" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.2.3.cmml">±</mo></msub><mo id="S2.Ex4.m1.2.2.1.1.1.1.1.1.1" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.2.2.1.1.1.1.1.1.3.2" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m1.2.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mi id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml">a</mi><mo id="S2.Ex4.m1.2.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.2.2.1.1.2" xref="S2.Ex4.m1.2.2.1.1.2.cmml">=</mo><mrow id="S2.Ex4.m1.2.2.1.1.3" xref="S2.Ex4.m1.2.2.1.1.3.cmml"><mn id="S2.Ex4.m1.2.2.1.1.3.2" xref="S2.Ex4.m1.2.2.1.1.3.2.cmml">1</mn><mo id="S2.Ex4.m1.2.2.1.1.3.1" xref="S2.Ex4.m1.2.2.1.1.3.1.cmml">−</mo><mstyle displaystyle="true" id="S2.Ex4.m1.2.2.1.1.3.3" xref="S2.Ex4.m1.2.2.1.1.3.3.cmml"><mfrac id="S2.Ex4.m1.2.2.1.1.3.3a" xref="S2.Ex4.m1.2.2.1.1.3.3.cmml"><mrow id="S2.Ex4.m1.2.2.1.1.3.3.2" xref="S2.Ex4.m1.2.2.1.1.3.3.2.cmml"><msup id="S2.Ex4.m1.2.2.1.1.3.3.2.2" xref="S2.Ex4.m1.2.2.1.1.3.3.2.2.cmml"><mi id="S2.Ex4.m1.2.2.1.1.3.3.2.2.2" 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xref="S2.Ex4.m1.2.2.1.1.3.2">1</cn><apply id="S2.Ex4.m1.2.2.1.1.3.3.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3"><divide id="S2.Ex4.m1.2.2.1.1.3.3.1.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3"></divide><apply id="S2.Ex4.m1.2.2.1.1.3.3.2.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2"><times id="S2.Ex4.m1.2.2.1.1.3.3.2.1.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2.1"></times><apply id="S2.Ex4.m1.2.2.1.1.3.3.2.2.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2.2"><csymbol cd="ambiguous" id="S2.Ex4.m1.2.2.1.1.3.3.2.2.1.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2.2">superscript</csymbol><ci id="S2.Ex4.m1.2.2.1.1.3.3.2.2.2.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2.2.2">𝑘</ci><cn id="S2.Ex4.m1.2.2.1.1.3.3.2.2.3.cmml" type="integer" xref="S2.Ex4.m1.2.2.1.1.3.3.2.2.3">2</cn></apply><apply id="S2.Ex4.m1.2.2.1.1.3.3.2.3.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S2.Ex4.m1.2.2.1.1.3.3.2.3.1.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2.3">superscript</csymbol><ci id="S2.Ex4.m1.2.2.1.1.3.3.2.3.2.cmml" xref="S2.Ex4.m1.2.2.1.1.3.3.2.3.2">𝑎</ci><cn id="S2.Ex4.m1.2.2.1.1.3.3.2.3.3.cmml" type="integer" xref="S2.Ex4.m1.2.2.1.1.3.3.2.3.3">2</cn></apply></apply><cn id="S2.Ex4.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S2.Ex4.m1.2.2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.2c">\displaystyle\text{Re}(g_{\pm}(a))=1-\frac{k^{2}a^{2}}{2},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.2d">Re ( italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) ) = 1 - divide start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.1"></minus><cn id="S2.Ex5.m1.2.2.1.1.3.2.5.2.2.cmml" type="integer" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.2">4</cn><apply id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3"><times id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.1.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.1"></times><apply id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2"><csymbol cd="ambiguous" id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2.1.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2">superscript</csymbol><ci id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2.2.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2.2">𝑘</ci><cn id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2.3.cmml" type="integer" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.2.3">2</cn></apply><apply id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3"><csymbol cd="ambiguous" id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3.1.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3">superscript</csymbol><ci id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3.2.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3.2">𝑎</ci><cn id="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3.3.cmml" type="integer" xref="S2.Ex5.m1.2.2.1.1.3.2.5.2.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m1.2c">\displaystyle\text{Im}(g_{\pm}(a))=\pm\frac{1}{2}ka\sqrt{4-k^{2}a^{2}}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.2d">Im ( italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) ) = ± divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_k italic_a square-root start_ARG 4 - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.3">As <math alttext="|g_{\pm}(a)|=1" class="ltx_Math" display="inline" id="S2.p6.3.m1.2"><semantics id="S2.p6.3.m1.2a"><mrow id="S2.p6.3.m1.2.2" 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xref="S2.p6.3.m1.2.2.2.cmml">=</mo><mn id="S2.p6.3.m1.2.2.3" xref="S2.p6.3.m1.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.3.m1.2b"><apply id="S2.p6.3.m1.2.2.cmml" xref="S2.p6.3.m1.2.2"><eq id="S2.p6.3.m1.2.2.2.cmml" xref="S2.p6.3.m1.2.2.2"></eq><apply id="S2.p6.3.m1.2.2.1.2.cmml" xref="S2.p6.3.m1.2.2.1.1"><abs id="S2.p6.3.m1.2.2.1.2.1.cmml" xref="S2.p6.3.m1.2.2.1.1.2"></abs><apply id="S2.p6.3.m1.2.2.1.1.1.cmml" xref="S2.p6.3.m1.2.2.1.1.1"><times id="S2.p6.3.m1.2.2.1.1.1.1.cmml" xref="S2.p6.3.m1.2.2.1.1.1.1"></times><apply id="S2.p6.3.m1.2.2.1.1.1.2.cmml" xref="S2.p6.3.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.p6.3.m1.2.2.1.1.1.2.1.cmml" xref="S2.p6.3.m1.2.2.1.1.1.2">subscript</csymbol><ci id="S2.p6.3.m1.2.2.1.1.1.2.2.cmml" xref="S2.p6.3.m1.2.2.1.1.1.2.2">𝑔</ci><csymbol cd="latexml" id="S2.p6.3.m1.2.2.1.1.1.2.3.cmml" xref="S2.p6.3.m1.2.2.1.1.1.2.3">plus-or-minus</csymbol></apply><ci id="S2.p6.3.m1.1.1.cmml" xref="S2.p6.3.m1.1.1">𝑎</ci></apply></apply><cn id="S2.p6.3.m1.2.2.3.cmml" type="integer" xref="S2.p6.3.m1.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.3.m1.2c">|g_{\pm}(a)|=1</annotation><annotation encoding="application/x-llamapun" id="S2.p6.3.m1.2d">| italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) | = 1</annotation></semantics></math>, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S2.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}&\varphi_{+}(a)=\left\{\begin{matrix}\arccos\left(1-k^{2}a^{2}/2% \right),&\text{if }k>0,\\[2.15277pt] 2\pi-\arccos\left(1-k^{2}a^{2}/2\right),&\text{if }k<0,\end{matrix}\right.\\[4% .30554pt] &\varphi_{-}(a)=\left\{\begin{matrix}2\pi-\arccos\left(1-k^{2}a^{2}/2\right),&% \text{if }k>0,\\[2.15277pt] \arccos\left(1-k^{2}a^{2}/2\right),&\text{if }k<0.\end{matrix}\right.\end{split}" class="ltx_math_unparsed" display="block" id="S2.E11.m1.16"><semantics id="S2.E11.m1.16a"><mtable columnspacing="0pt" displaystyle="true" id="S2.E11.m1.16.16" rowspacing="0pt"><mtr id="S2.E11.m1.16.16a"><mtd id="S2.E11.m1.16.16b"></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.E11.m1.16.16c"><mrow id="S2.E11.m1.9.9.9.8.8"><msub id="S2.E11.m1.9.9.9.8.8.9"><mi id="S2.E11.m1.3.3.3.2.2.2">φ</mi><mo id="S2.E11.m1.4.4.4.3.3.3.1">+</mo></msub><mrow id="S2.E11.m1.9.9.9.8.8.10"><mo id="S2.E11.m1.5.5.5.4.4.4" stretchy="false">(</mo><mi id="S2.E11.m1.6.6.6.5.5.5">a</mi><mo id="S2.E11.m1.7.7.7.6.6.6" stretchy="false">)</mo></mrow><mo id="S2.E11.m1.8.8.8.7.7.7">=</mo><mrow id="S2.E11.m1.9.9.9.8.8.11"><mo id="S2.E11.m1.9.9.9.8.8.8">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S2.E11.m1.1.1.1.1.1.1.1.1" rowspacing="0pt"><mtr id="S2.E11.m1.1.1.1.1.1.1.1.1a"><mtd id="S2.E11.m1.1.1.1.1.1.1.1.1b"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1"><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.1.1.1.1.1">arccos</mi><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1a">⁡</mo><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1"><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.2">(</mo><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1"><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.2">1</mn><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1">−</mo><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2"><msup id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2.2"><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2.2.2">k</mi><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2.2.3">2</mn></msup><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2.1">⁢</mo><msup id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2.3"><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2.3.2">a</mi><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.2.3.3">2</mn></msup></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.1">/</mo><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3.3">2</mn></mrow></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.1.1.1.3">)</mo></mrow></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.2.2.2.2.2.2">,</mo></mrow></mtd><mtd id="S2.E11.m1.1.1.1.1.1.1.1.1c"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.1"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.2"><mtext id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.2.2">if </mtext><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.2.1">⁢</mo><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.2.3">k</mi></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.1">></mo><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.3">0</mn></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.3.3.3.1.1.2">,</mo></mrow></mtd></mtr><mtr id="S2.E11.m1.1.1.1.1.1.1.1.1d"><mtd id="S2.E11.m1.1.1.1.1.1.1.1.1e"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.3"><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.3.2">2</mn><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.3.1">⁢</mo><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.3.3">π</mi></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.2">−</mo><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1"><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.4.4.1.1.1">arccos</mi><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1a">⁡</mo><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1"><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.2">(</mo><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1"><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.2">1</mn><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.1">−</mo><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2"><msup id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2.2"><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2.2.2">k</mi><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2.2.3">2</mn></msup><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2.1">⁢</mo><msup id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2.3"><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2.3.2">a</mi><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.2.3.3">2</mn></msup></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.1">/</mo><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1.3.3">2</mn></mrow></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3">)</mo></mrow></mrow></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.5.5.2.2.2.2">,</mo></mrow></mtd><mtd id="S2.E11.m1.1.1.1.1.1.1.1.1f"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.1"><mrow id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.1.2"><mtext id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.1.2.2">if </mtext><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.1.2.1">⁢</mo><mi id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.1.2.3">k</mi></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.1.1"><</mo><mn id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.1.3">0</mn></mrow><mo id="S2.E11.m1.1.1.1.1.1.1.1.1.6.6.3.1.1.2">,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mtd></mtr><mtr id="S2.E11.m1.16.16d"><mtd id="S2.E11.m1.16.16e"></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.E11.m1.16.16f"><mrow id="S2.E11.m1.16.16.16.8.8"><msub id="S2.E11.m1.16.16.16.8.8.9"><mi id="S2.E11.m1.10.10.10.2.2.2">φ</mi><mo id="S2.E11.m1.11.11.11.3.3.3.1">−</mo></msub><mrow id="S2.E11.m1.16.16.16.8.8.10"><mo id="S2.E11.m1.12.12.12.4.4.4" stretchy="false">(</mo><mi id="S2.E11.m1.13.13.13.5.5.5">a</mi><mo id="S2.E11.m1.14.14.14.6.6.6" stretchy="false">)</mo></mrow><mo id="S2.E11.m1.15.15.15.7.7.7">=</mo><mrow id="S2.E11.m1.16.16.16.8.8.11"><mo id="S2.E11.m1.16.16.16.8.8.8">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S2.E11.m1.2.2.2.1.1.1.1.1" rowspacing="0pt"><mtr id="S2.E11.m1.2.2.2.1.1.1.1.1a"><mtd id="S2.E11.m1.2.2.2.1.1.1.1.1b"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.3"><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.3.2">2</mn><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.3.1">⁢</mo><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.3.3">π</mi></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.2">−</mo><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1"><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.1.1.1.1.1">arccos</mi><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1a">⁡</mo><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1"><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.2">(</mo><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1"><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.2">1</mn><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.1">−</mo><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2"><msup id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2.2"><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2.2.2">k</mi><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2.2.3">2</mn></msup><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2.1">⁢</mo><msup id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2.3"><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2.3.2">a</mi><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.2.3.3">2</mn></msup></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.1">/</mo><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.1.3.3">2</mn></mrow></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.1.1.1.1.3">)</mo></mrow></mrow></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.2.2.2.2.2.2">,</mo></mrow></mtd><mtd id="S2.E11.m1.2.2.2.1.1.1.1.1c"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.1"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.1.2"><mtext id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.1.2.2">if </mtext><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.1.2.1">⁢</mo><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.1.2.3">k</mi></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.1.1">></mo><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.1.3">0</mn></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.3.3.3.1.1.2">,</mo></mrow></mtd></mtr><mtr id="S2.E11.m1.2.2.2.1.1.1.1.1d"><mtd id="S2.E11.m1.2.2.2.1.1.1.1.1e"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1"><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.4.4.1.1.1">arccos</mi><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1a">⁡</mo><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1"><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.2">(</mo><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1"><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.2">1</mn><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.1">−</mo><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2"><msup id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2.2"><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2.2.2">k</mi><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2.2.3">2</mn></msup><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2.1">⁢</mo><msup id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2.3"><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2.3.2">a</mi><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.2.3.3">2</mn></msup></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.1">/</mo><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.1.3.3">2</mn></mrow></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.1.1.1.3">)</mo></mrow></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.5.5.2.2.2.2">,</mo></mrow></mtd><mtd id="S2.E11.m1.2.2.2.1.1.1.1.1f"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.1"><mrow id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.1.2"><mtext id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.1.2.2">if </mtext><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.1.2.1">⁢</mo><mi id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.1.2.3">k</mi></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.1.1"><</mo><mn id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.1.3">0</mn></mrow><mo id="S2.E11.m1.2.2.2.1.1.1.1.1.6.6.3.1.1.2" lspace="0em">.</mo></mrow></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.E11.m1.16b">\begin{split}&\varphi_{+}(a)=\left\{\begin{matrix}\arccos\left(1-k^{2}a^{2}/2% \right),&\text{if }k>0,\\[2.15277pt] 2\pi-\arccos\left(1-k^{2}a^{2}/2\right),&\text{if }k<0,\end{matrix}\right.\\[4% .30554pt] &\varphi_{-}(a)=\left\{\begin{matrix}2\pi-\arccos\left(1-k^{2}a^{2}/2\right),&% \text{if }k>0,\\[2.15277pt] \arccos\left(1-k^{2}a^{2}/2\right),&\text{if }k<0.\end{matrix}\right.\end{split}</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m1.16c">start_ROW start_CELL end_CELL start_CELL italic_φ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) = { start_ARG start_ROW start_CELL roman_arccos ( 1 - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ) , end_CELL start_CELL if italic_k > 0 , end_CELL end_ROW start_ROW start_CELL 2 italic_π - roman_arccos ( 1 - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ) , end_CELL start_CELL if italic_k < 0 , end_CELL end_ROW end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_φ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) = { start_ARG start_ROW start_CELL 2 italic_π - roman_arccos ( 1 - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ) , end_CELL start_CELL if italic_k > 0 , end_CELL end_ROW start_ROW start_CELL roman_arccos ( 1 - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ) , end_CELL start_CELL if italic_k < 0 . end_CELL end_ROW end_ARG end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.5">By substitution of these equations into the secular equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E8" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">8</span></a>) we see that <math alttext="N\varphi_{+}(a)-N\varphi_{-}(a)" class="ltx_Math" display="inline" id="S2.p6.4.m1.2"><semantics id="S2.p6.4.m1.2a"><mrow id="S2.p6.4.m1.2.3" xref="S2.p6.4.m1.2.3.cmml"><mrow id="S2.p6.4.m1.2.3.2" xref="S2.p6.4.m1.2.3.2.cmml"><mi id="S2.p6.4.m1.2.3.2.2" xref="S2.p6.4.m1.2.3.2.2.cmml">N</mi><mo id="S2.p6.4.m1.2.3.2.1" xref="S2.p6.4.m1.2.3.2.1.cmml">⁢</mo><msub id="S2.p6.4.m1.2.3.2.3" xref="S2.p6.4.m1.2.3.2.3.cmml"><mi id="S2.p6.4.m1.2.3.2.3.2" xref="S2.p6.4.m1.2.3.2.3.2.cmml">φ</mi><mo id="S2.p6.4.m1.2.3.2.3.3" xref="S2.p6.4.m1.2.3.2.3.3.cmml">+</mo></msub><mo id="S2.p6.4.m1.2.3.2.1a" xref="S2.p6.4.m1.2.3.2.1.cmml">⁢</mo><mrow id="S2.p6.4.m1.2.3.2.4.2" xref="S2.p6.4.m1.2.3.2.cmml"><mo id="S2.p6.4.m1.2.3.2.4.2.1" stretchy="false" xref="S2.p6.4.m1.2.3.2.cmml">(</mo><mi id="S2.p6.4.m1.1.1" xref="S2.p6.4.m1.1.1.cmml">a</mi><mo id="S2.p6.4.m1.2.3.2.4.2.2" stretchy="false" xref="S2.p6.4.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.p6.4.m1.2.3.1" xref="S2.p6.4.m1.2.3.1.cmml">−</mo><mrow id="S2.p6.4.m1.2.3.3" xref="S2.p6.4.m1.2.3.3.cmml"><mi id="S2.p6.4.m1.2.3.3.2" xref="S2.p6.4.m1.2.3.3.2.cmml">N</mi><mo id="S2.p6.4.m1.2.3.3.1" xref="S2.p6.4.m1.2.3.3.1.cmml">⁢</mo><msub id="S2.p6.4.m1.2.3.3.3" xref="S2.p6.4.m1.2.3.3.3.cmml"><mi id="S2.p6.4.m1.2.3.3.3.2" xref="S2.p6.4.m1.2.3.3.3.2.cmml">φ</mi><mo id="S2.p6.4.m1.2.3.3.3.3" xref="S2.p6.4.m1.2.3.3.3.3.cmml">−</mo></msub><mo id="S2.p6.4.m1.2.3.3.1a" xref="S2.p6.4.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.p6.4.m1.2.3.3.4.2" xref="S2.p6.4.m1.2.3.3.cmml"><mo id="S2.p6.4.m1.2.3.3.4.2.1" stretchy="false" xref="S2.p6.4.m1.2.3.3.cmml">(</mo><mi id="S2.p6.4.m1.2.2" xref="S2.p6.4.m1.2.2.cmml">a</mi><mo id="S2.p6.4.m1.2.3.3.4.2.2" stretchy="false" xref="S2.p6.4.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.4.m1.2b"><apply id="S2.p6.4.m1.2.3.cmml" xref="S2.p6.4.m1.2.3"><minus id="S2.p6.4.m1.2.3.1.cmml" xref="S2.p6.4.m1.2.3.1"></minus><apply id="S2.p6.4.m1.2.3.2.cmml" xref="S2.p6.4.m1.2.3.2"><times id="S2.p6.4.m1.2.3.2.1.cmml" xref="S2.p6.4.m1.2.3.2.1"></times><ci id="S2.p6.4.m1.2.3.2.2.cmml" xref="S2.p6.4.m1.2.3.2.2">𝑁</ci><apply id="S2.p6.4.m1.2.3.2.3.cmml" xref="S2.p6.4.m1.2.3.2.3"><csymbol cd="ambiguous" id="S2.p6.4.m1.2.3.2.3.1.cmml" xref="S2.p6.4.m1.2.3.2.3">subscript</csymbol><ci id="S2.p6.4.m1.2.3.2.3.2.cmml" xref="S2.p6.4.m1.2.3.2.3.2">𝜑</ci><plus id="S2.p6.4.m1.2.3.2.3.3.cmml" xref="S2.p6.4.m1.2.3.2.3.3"></plus></apply><ci id="S2.p6.4.m1.1.1.cmml" xref="S2.p6.4.m1.1.1">𝑎</ci></apply><apply id="S2.p6.4.m1.2.3.3.cmml" xref="S2.p6.4.m1.2.3.3"><times id="S2.p6.4.m1.2.3.3.1.cmml" xref="S2.p6.4.m1.2.3.3.1"></times><ci id="S2.p6.4.m1.2.3.3.2.cmml" xref="S2.p6.4.m1.2.3.3.2">𝑁</ci><apply id="S2.p6.4.m1.2.3.3.3.cmml" xref="S2.p6.4.m1.2.3.3.3"><csymbol cd="ambiguous" id="S2.p6.4.m1.2.3.3.3.1.cmml" xref="S2.p6.4.m1.2.3.3.3">subscript</csymbol><ci id="S2.p6.4.m1.2.3.3.3.2.cmml" xref="S2.p6.4.m1.2.3.3.3.2">𝜑</ci><minus id="S2.p6.4.m1.2.3.3.3.3.cmml" xref="S2.p6.4.m1.2.3.3.3.3"></minus></apply><ci id="S2.p6.4.m1.2.2.cmml" xref="S2.p6.4.m1.2.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.4.m1.2c">N\varphi_{+}(a)-N\varphi_{-}(a)</annotation><annotation encoding="application/x-llamapun" id="S2.p6.4.m1.2d">italic_N italic_φ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) - italic_N italic_φ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math> equals a multiple of <math alttext="2\pi" class="ltx_Math" display="inline" id="S2.p6.5.m2.1"><semantics id="S2.p6.5.m2.1a"><mrow id="S2.p6.5.m2.1.1" xref="S2.p6.5.m2.1.1.cmml"><mn id="S2.p6.5.m2.1.1.2" xref="S2.p6.5.m2.1.1.2.cmml">2</mn><mo id="S2.p6.5.m2.1.1.1" xref="S2.p6.5.m2.1.1.1.cmml">⁢</mo><mi id="S2.p6.5.m2.1.1.3" xref="S2.p6.5.m2.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.5.m2.1b"><apply id="S2.p6.5.m2.1.1.cmml" xref="S2.p6.5.m2.1.1"><times id="S2.p6.5.m2.1.1.1.cmml" xref="S2.p6.5.m2.1.1.1"></times><cn id="S2.p6.5.m2.1.1.2.cmml" type="integer" xref="S2.p6.5.m2.1.1.2">2</cn><ci id="S2.p6.5.m2.1.1.3.cmml" xref="S2.p6.5.m2.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.5.m2.1c">2\pi</annotation><annotation encoding="application/x-llamapun" id="S2.p6.5.m2.1d">2 italic_π</annotation></semantics></math>, which gives by (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E11" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">11</span></a>)</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\arccos\left(1-k^{2}a^{2}/2\right)=\frac{\pi m}{N}," class="ltx_Math" display="block" id="S2.Ex6.m1.2"><semantics id="S2.Ex6.m1.2a"><mrow id="S2.Ex6.m1.2.2.1" xref="S2.Ex6.m1.2.2.1.1.cmml"><mrow id="S2.Ex6.m1.2.2.1.1" xref="S2.Ex6.m1.2.2.1.1.cmml"><mrow id="S2.Ex6.m1.2.2.1.1.1.1" xref="S2.Ex6.m1.2.2.1.1.1.2.cmml"><mi id="S2.Ex6.m1.1.1" xref="S2.Ex6.m1.1.1.cmml">arccos</mi><mo id="S2.Ex6.m1.2.2.1.1.1.1a" xref="S2.Ex6.m1.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S2.Ex6.m1.2.2.1.1.1.1.1" xref="S2.Ex6.m1.2.2.1.1.1.2.cmml"><mo id="S2.Ex6.m1.2.2.1.1.1.1.1.2" xref="S2.Ex6.m1.2.2.1.1.1.2.cmml">(</mo><mrow id="S2.Ex6.m1.2.2.1.1.1.1.1.1" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.cmml"><mn id="S2.Ex6.m1.2.2.1.1.1.1.1.1.2" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.Ex6.m1.2.2.1.1.1.1.1.1.1" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.cmml"><mrow id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.cmml"><msup id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2.cmml"><mi id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2.2" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2.2.cmml">k</mi><mn id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2.3" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2.3.cmml">2</mn></msup><mo id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.1" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><msup id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.cmml"><mi id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.2" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.2.cmml">a</mi><mn id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.3" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.3.cmml">2</mn></msup></mrow><mo id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.1" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.3" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.3.cmml">2</mn></mrow></mrow><mo id="S2.Ex6.m1.2.2.1.1.1.1.1.3" xref="S2.Ex6.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex6.m1.2.2.1.1.2" xref="S2.Ex6.m1.2.2.1.1.2.cmml">=</mo><mfrac id="S2.Ex6.m1.2.2.1.1.3" xref="S2.Ex6.m1.2.2.1.1.3.cmml"><mrow id="S2.Ex6.m1.2.2.1.1.3.2" xref="S2.Ex6.m1.2.2.1.1.3.2.cmml"><mi id="S2.Ex6.m1.2.2.1.1.3.2.2" xref="S2.Ex6.m1.2.2.1.1.3.2.2.cmml">π</mi><mo id="S2.Ex6.m1.2.2.1.1.3.2.1" xref="S2.Ex6.m1.2.2.1.1.3.2.1.cmml">⁢</mo><mi id="S2.Ex6.m1.2.2.1.1.3.2.3" xref="S2.Ex6.m1.2.2.1.1.3.2.3.cmml">m</mi></mrow><mi id="S2.Ex6.m1.2.2.1.1.3.3" xref="S2.Ex6.m1.2.2.1.1.3.3.cmml">N</mi></mfrac></mrow><mo id="S2.Ex6.m1.2.2.1.2" xref="S2.Ex6.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex6.m1.2b"><apply id="S2.Ex6.m1.2.2.1.1.cmml" xref="S2.Ex6.m1.2.2.1"><eq id="S2.Ex6.m1.2.2.1.1.2.cmml" xref="S2.Ex6.m1.2.2.1.1.2"></eq><apply id="S2.Ex6.m1.2.2.1.1.1.2.cmml" xref="S2.Ex6.m1.2.2.1.1.1.1"><arccos 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id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2.3.cmml" type="integer" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.2.3">2</cn></apply><apply id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.cmml" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.1.cmml" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3">superscript</csymbol><ci id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.2.cmml" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.2">𝑎</ci><cn id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.2.3.3">2</cn></apply></apply><cn id="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.Ex6.m1.2.2.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply><apply id="S2.Ex6.m1.2.2.1.1.3.cmml" xref="S2.Ex6.m1.2.2.1.1.3"><divide id="S2.Ex6.m1.2.2.1.1.3.1.cmml" xref="S2.Ex6.m1.2.2.1.1.3"></divide><apply id="S2.Ex6.m1.2.2.1.1.3.2.cmml" xref="S2.Ex6.m1.2.2.1.1.3.2"><times id="S2.Ex6.m1.2.2.1.1.3.2.1.cmml" xref="S2.Ex6.m1.2.2.1.1.3.2.1"></times><ci id="S2.Ex6.m1.2.2.1.1.3.2.2.cmml" xref="S2.Ex6.m1.2.2.1.1.3.2.2">𝜋</ci><ci id="S2.Ex6.m1.2.2.1.1.3.2.3.cmml" xref="S2.Ex6.m1.2.2.1.1.3.2.3">𝑚</ci></apply><ci id="S2.Ex6.m1.2.2.1.1.3.3.cmml" xref="S2.Ex6.m1.2.2.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m1.2c">\arccos\left(1-k^{2}a^{2}/2\right)=\frac{\pi m}{N},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m1.2d">roman_arccos ( 1 - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ) = divide start_ARG italic_π italic_m end_ARG start_ARG italic_N end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.8">where <math alttext="m" class="ltx_Math" display="inline" id="S2.p6.6.m1.1"><semantics id="S2.p6.6.m1.1a"><mi id="S2.p6.6.m1.1.1" xref="S2.p6.6.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.p6.6.m1.1b"><ci id="S2.p6.6.m1.1.1.cmml" xref="S2.p6.6.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.6.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p6.6.m1.1d">italic_m</annotation></semantics></math> is an integer. Hence for every integer <math alttext="m" class="ltx_Math" display="inline" id="S2.p6.7.m2.1"><semantics id="S2.p6.7.m2.1a"><mi id="S2.p6.7.m2.1.1" xref="S2.p6.7.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.p6.7.m2.1b"><ci id="S2.p6.7.m2.1.1.cmml" xref="S2.p6.7.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.7.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p6.7.m2.1d">italic_m</annotation></semantics></math> we obtain a solution <math alttext="k_{m}" class="ltx_Math" display="inline" id="S2.p6.8.m3.1"><semantics id="S2.p6.8.m3.1a"><msub id="S2.p6.8.m3.1.1" xref="S2.p6.8.m3.1.1.cmml"><mi id="S2.p6.8.m3.1.1.2" xref="S2.p6.8.m3.1.1.2.cmml">k</mi><mi id="S2.p6.8.m3.1.1.3" xref="S2.p6.8.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p6.8.m3.1b"><apply id="S2.p6.8.m3.1.1.cmml" xref="S2.p6.8.m3.1.1"><csymbol cd="ambiguous" id="S2.p6.8.m3.1.1.1.cmml" xref="S2.p6.8.m3.1.1">subscript</csymbol><ci id="S2.p6.8.m3.1.1.2.cmml" xref="S2.p6.8.m3.1.1.2">𝑘</ci><ci id="S2.p6.8.m3.1.1.3.cmml" xref="S2.p6.8.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.8.m3.1c">k_{m}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.8.m3.1d">italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx9"> <tbody id="S2.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle k_{m}=\pm\frac{1}{a}\sqrt{2-2\cos\left(\frac{\pi m}{N}\right)}." class="ltx_Math" display="inline" id="S2.E12.m1.3"><semantics id="S2.E12.m1.3a"><mrow id="S2.E12.m1.3.3.1" xref="S2.E12.m1.3.3.1.1.cmml"><mrow id="S2.E12.m1.3.3.1.1" xref="S2.E12.m1.3.3.1.1.cmml"><msub id="S2.E12.m1.3.3.1.1.2" xref="S2.E12.m1.3.3.1.1.2.cmml"><mi id="S2.E12.m1.3.3.1.1.2.2" xref="S2.E12.m1.3.3.1.1.2.2.cmml">k</mi><mi id="S2.E12.m1.3.3.1.1.2.3" xref="S2.E12.m1.3.3.1.1.2.3.cmml">m</mi></msub><mo id="S2.E12.m1.3.3.1.1.1" xref="S2.E12.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E12.m1.3.3.1.1.3" xref="S2.E12.m1.3.3.1.1.3.cmml"><mo id="S2.E12.m1.3.3.1.1.3a" xref="S2.E12.m1.3.3.1.1.3.cmml">±</mo><mrow id="S2.E12.m1.3.3.1.1.3.2" xref="S2.E12.m1.3.3.1.1.3.2.cmml"><mstyle displaystyle="true" id="S2.E12.m1.3.3.1.1.3.2.2" xref="S2.E12.m1.3.3.1.1.3.2.2.cmml"><mfrac id="S2.E12.m1.3.3.1.1.3.2.2a" xref="S2.E12.m1.3.3.1.1.3.2.2.cmml"><mn id="S2.E12.m1.3.3.1.1.3.2.2.2" xref="S2.E12.m1.3.3.1.1.3.2.2.2.cmml">1</mn><mi id="S2.E12.m1.3.3.1.1.3.2.2.3" xref="S2.E12.m1.3.3.1.1.3.2.2.3.cmml">a</mi></mfrac></mstyle><mo id="S2.E12.m1.3.3.1.1.3.2.1" xref="S2.E12.m1.3.3.1.1.3.2.1.cmml">⁢</mo><msqrt id="S2.E12.m1.2.2" xref="S2.E12.m1.2.2.cmml"><mrow id="S2.E12.m1.2.2.2" xref="S2.E12.m1.2.2.2.cmml"><mn id="S2.E12.m1.2.2.2.4" xref="S2.E12.m1.2.2.2.4.cmml">2</mn><mo id="S2.E12.m1.2.2.2.3" xref="S2.E12.m1.2.2.2.3.cmml">−</mo><mrow id="S2.E12.m1.2.2.2.5" xref="S2.E12.m1.2.2.2.5.cmml"><mn id="S2.E12.m1.2.2.2.5.2" xref="S2.E12.m1.2.2.2.5.2.cmml">2</mn><mo id="S2.E12.m1.2.2.2.5.1" lspace="0.167em" xref="S2.E12.m1.2.2.2.5.1.cmml">⁢</mo><mrow id="S2.E12.m1.2.2.2.5.3.2" xref="S2.E12.m1.2.2.2.5.3.1.cmml"><mi id="S2.E12.m1.1.1.1.1" xref="S2.E12.m1.1.1.1.1.cmml">cos</mi><mo id="S2.E12.m1.2.2.2.5.3.2a" xref="S2.E12.m1.2.2.2.5.3.1.cmml">⁡</mo><mrow id="S2.E12.m1.2.2.2.5.3.2.1" xref="S2.E12.m1.2.2.2.5.3.1.cmml"><mo id="S2.E12.m1.2.2.2.5.3.2.1.1" xref="S2.E12.m1.2.2.2.5.3.1.cmml">(</mo><mstyle displaystyle="true" id="S2.E12.m1.2.2.2.2" xref="S2.E12.m1.2.2.2.2.cmml"><mfrac id="S2.E12.m1.2.2.2.2a" xref="S2.E12.m1.2.2.2.2.cmml"><mrow id="S2.E12.m1.2.2.2.2.2" xref="S2.E12.m1.2.2.2.2.2.cmml"><mi id="S2.E12.m1.2.2.2.2.2.2" xref="S2.E12.m1.2.2.2.2.2.2.cmml">π</mi><mo id="S2.E12.m1.2.2.2.2.2.1" xref="S2.E12.m1.2.2.2.2.2.1.cmml">⁢</mo><mi id="S2.E12.m1.2.2.2.2.2.3" xref="S2.E12.m1.2.2.2.2.2.3.cmml">m</mi></mrow><mi id="S2.E12.m1.2.2.2.2.3" xref="S2.E12.m1.2.2.2.2.3.cmml">N</mi></mfrac></mstyle><mo id="S2.E12.m1.2.2.2.5.3.2.1.2" xref="S2.E12.m1.2.2.2.5.3.1.cmml">)</mo></mrow></mrow></mrow></mrow></msqrt></mrow></mrow></mrow><mo id="S2.E12.m1.3.3.1.2" lspace="0em" xref="S2.E12.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E12.m1.3b"><apply id="S2.E12.m1.3.3.1.1.cmml" xref="S2.E12.m1.3.3.1"><eq id="S2.E12.m1.3.3.1.1.1.cmml" xref="S2.E12.m1.3.3.1.1.1"></eq><apply id="S2.E12.m1.3.3.1.1.2.cmml" xref="S2.E12.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.E12.m1.3.3.1.1.2.1.cmml" xref="S2.E12.m1.3.3.1.1.2">subscript</csymbol><ci id="S2.E12.m1.3.3.1.1.2.2.cmml" xref="S2.E12.m1.3.3.1.1.2.2">𝑘</ci><ci id="S2.E12.m1.3.3.1.1.2.3.cmml" xref="S2.E12.m1.3.3.1.1.2.3">𝑚</ci></apply><apply id="S2.E12.m1.3.3.1.1.3.cmml" xref="S2.E12.m1.3.3.1.1.3"><csymbol cd="latexml" id="S2.E12.m1.3.3.1.1.3.1.cmml" xref="S2.E12.m1.3.3.1.1.3">plus-or-minus</csymbol><apply id="S2.E12.m1.3.3.1.1.3.2.cmml" xref="S2.E12.m1.3.3.1.1.3.2"><times id="S2.E12.m1.3.3.1.1.3.2.1.cmml" xref="S2.E12.m1.3.3.1.1.3.2.1"></times><apply id="S2.E12.m1.3.3.1.1.3.2.2.cmml" xref="S2.E12.m1.3.3.1.1.3.2.2"><divide id="S2.E12.m1.3.3.1.1.3.2.2.1.cmml" xref="S2.E12.m1.3.3.1.1.3.2.2"></divide><cn id="S2.E12.m1.3.3.1.1.3.2.2.2.cmml" type="integer" xref="S2.E12.m1.3.3.1.1.3.2.2.2">1</cn><ci id="S2.E12.m1.3.3.1.1.3.2.2.3.cmml" xref="S2.E12.m1.3.3.1.1.3.2.2.3">𝑎</ci></apply><apply id="S2.E12.m1.2.2.cmml" xref="S2.E12.m1.2.2"><root id="S2.E12.m1.2.2a.cmml" xref="S2.E12.m1.2.2"></root><apply id="S2.E12.m1.2.2.2.cmml" xref="S2.E12.m1.2.2.2"><minus id="S2.E12.m1.2.2.2.3.cmml" xref="S2.E12.m1.2.2.2.3"></minus><cn id="S2.E12.m1.2.2.2.4.cmml" type="integer" xref="S2.E12.m1.2.2.2.4">2</cn><apply id="S2.E12.m1.2.2.2.5.cmml" xref="S2.E12.m1.2.2.2.5"><times id="S2.E12.m1.2.2.2.5.1.cmml" xref="S2.E12.m1.2.2.2.5.1"></times><cn id="S2.E12.m1.2.2.2.5.2.cmml" type="integer" xref="S2.E12.m1.2.2.2.5.2">2</cn><apply id="S2.E12.m1.2.2.2.5.3.1.cmml" xref="S2.E12.m1.2.2.2.5.3.2"><cos id="S2.E12.m1.1.1.1.1.cmml" xref="S2.E12.m1.1.1.1.1"></cos><apply id="S2.E12.m1.2.2.2.2.cmml" xref="S2.E12.m1.2.2.2.2"><divide id="S2.E12.m1.2.2.2.2.1.cmml" xref="S2.E12.m1.2.2.2.2"></divide><apply id="S2.E12.m1.2.2.2.2.2.cmml" xref="S2.E12.m1.2.2.2.2.2"><times id="S2.E12.m1.2.2.2.2.2.1.cmml" xref="S2.E12.m1.2.2.2.2.2.1"></times><ci id="S2.E12.m1.2.2.2.2.2.2.cmml" xref="S2.E12.m1.2.2.2.2.2.2">𝜋</ci><ci id="S2.E12.m1.2.2.2.2.2.3.cmml" xref="S2.E12.m1.2.2.2.2.2.3">𝑚</ci></apply><ci id="S2.E12.m1.2.2.2.2.3.cmml" xref="S2.E12.m1.2.2.2.2.3">𝑁</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E12.m1.3c">\displaystyle k_{m}=\pm\frac{1}{a}\sqrt{2-2\cos\left(\frac{\pi m}{N}\right)}.</annotation><annotation encoding="application/x-llamapun" id="S2.E12.m1.3d">italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ± divide start_ARG 1 end_ARG start_ARG italic_a end_ARG square-root start_ARG 2 - 2 roman_cos ( divide start_ARG italic_π italic_m end_ARG start_ARG italic_N end_ARG ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.14">That is, equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E2" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a>) equipped with Dirichlet boundary conditions has solutions only if <math alttext="k\in\{k_{m}\,:\,m\in\mathbb{Z}\}" class="ltx_Math" display="inline" id="S2.p6.9.m1.2"><semantics id="S2.p6.9.m1.2a"><mrow id="S2.p6.9.m1.2.2" xref="S2.p6.9.m1.2.2.cmml"><mi id="S2.p6.9.m1.2.2.4" xref="S2.p6.9.m1.2.2.4.cmml">k</mi><mo id="S2.p6.9.m1.2.2.3" xref="S2.p6.9.m1.2.2.3.cmml">∈</mo><mrow id="S2.p6.9.m1.2.2.2.2" xref="S2.p6.9.m1.2.2.2.3.cmml"><mo id="S2.p6.9.m1.2.2.2.2.3" stretchy="false" xref="S2.p6.9.m1.2.2.2.3.1.cmml">{</mo><msub id="S2.p6.9.m1.1.1.1.1.1" xref="S2.p6.9.m1.1.1.1.1.1.cmml"><mi id="S2.p6.9.m1.1.1.1.1.1.2" xref="S2.p6.9.m1.1.1.1.1.1.2.cmml">k</mi><mi id="S2.p6.9.m1.1.1.1.1.1.3" xref="S2.p6.9.m1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S2.p6.9.m1.2.2.2.2.4" lspace="0.278em" rspace="0.448em" xref="S2.p6.9.m1.2.2.2.3.1.cmml">:</mo><mrow id="S2.p6.9.m1.2.2.2.2.2" xref="S2.p6.9.m1.2.2.2.2.2.cmml"><mi id="S2.p6.9.m1.2.2.2.2.2.2" xref="S2.p6.9.m1.2.2.2.2.2.2.cmml">m</mi><mo id="S2.p6.9.m1.2.2.2.2.2.1" xref="S2.p6.9.m1.2.2.2.2.2.1.cmml">∈</mo><mi id="S2.p6.9.m1.2.2.2.2.2.3" xref="S2.p6.9.m1.2.2.2.2.2.3.cmml">ℤ</mi></mrow><mo id="S2.p6.9.m1.2.2.2.2.5" stretchy="false" xref="S2.p6.9.m1.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.9.m1.2b"><apply id="S2.p6.9.m1.2.2.cmml" xref="S2.p6.9.m1.2.2"><in id="S2.p6.9.m1.2.2.3.cmml" xref="S2.p6.9.m1.2.2.3"></in><ci id="S2.p6.9.m1.2.2.4.cmml" xref="S2.p6.9.m1.2.2.4">𝑘</ci><apply id="S2.p6.9.m1.2.2.2.3.cmml" xref="S2.p6.9.m1.2.2.2.2"><csymbol cd="latexml" id="S2.p6.9.m1.2.2.2.3.1.cmml" xref="S2.p6.9.m1.2.2.2.2.3">conditional-set</csymbol><apply id="S2.p6.9.m1.1.1.1.1.1.cmml" xref="S2.p6.9.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p6.9.m1.1.1.1.1.1.1.cmml" xref="S2.p6.9.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.p6.9.m1.1.1.1.1.1.2.cmml" xref="S2.p6.9.m1.1.1.1.1.1.2">𝑘</ci><ci id="S2.p6.9.m1.1.1.1.1.1.3.cmml" xref="S2.p6.9.m1.1.1.1.1.1.3">𝑚</ci></apply><apply id="S2.p6.9.m1.2.2.2.2.2.cmml" xref="S2.p6.9.m1.2.2.2.2.2"><in id="S2.p6.9.m1.2.2.2.2.2.1.cmml" xref="S2.p6.9.m1.2.2.2.2.2.1"></in><ci id="S2.p6.9.m1.2.2.2.2.2.2.cmml" xref="S2.p6.9.m1.2.2.2.2.2.2">𝑚</ci><ci id="S2.p6.9.m1.2.2.2.2.2.3.cmml" xref="S2.p6.9.m1.2.2.2.2.2.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.9.m1.2c">k\in\{k_{m}\,:\,m\in\mathbb{Z}\}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.9.m1.2d">italic_k ∈ { italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_m ∈ blackboard_Z }</annotation></semantics></math>. Obviously, the <math alttext="k_{m}" class="ltx_Math" display="inline" id="S2.p6.10.m2.1"><semantics id="S2.p6.10.m2.1a"><msub id="S2.p6.10.m2.1.1" xref="S2.p6.10.m2.1.1.cmml"><mi id="S2.p6.10.m2.1.1.2" xref="S2.p6.10.m2.1.1.2.cmml">k</mi><mi id="S2.p6.10.m2.1.1.3" xref="S2.p6.10.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p6.10.m2.1b"><apply id="S2.p6.10.m2.1.1.cmml" xref="S2.p6.10.m2.1.1"><csymbol cd="ambiguous" id="S2.p6.10.m2.1.1.1.cmml" xref="S2.p6.10.m2.1.1">subscript</csymbol><ci id="S2.p6.10.m2.1.1.2.cmml" xref="S2.p6.10.m2.1.1.2">𝑘</ci><ci id="S2.p6.10.m2.1.1.3.cmml" xref="S2.p6.10.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.10.m2.1c">k_{m}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.10.m2.1d">italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> depend also on the step size <math alttext="a" class="ltx_Math" display="inline" id="S2.p6.11.m3.1"><semantics id="S2.p6.11.m3.1a"><mi id="S2.p6.11.m3.1.1" xref="S2.p6.11.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.p6.11.m3.1b"><ci id="S2.p6.11.m3.1.1.cmml" xref="S2.p6.11.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.11.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.p6.11.m3.1d">italic_a</annotation></semantics></math>. If the step size <math alttext="a" class="ltx_Math" display="inline" id="S2.p6.12.m4.1"><semantics id="S2.p6.12.m4.1a"><mi id="S2.p6.12.m4.1.1" xref="S2.p6.12.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.p6.12.m4.1b"><ci id="S2.p6.12.m4.1.1.cmml" xref="S2.p6.12.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.12.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.p6.12.m4.1d">italic_a</annotation></semantics></math> tends to zero, then <math alttext="k_{m}" class="ltx_Math" display="inline" id="S2.p6.13.m5.1"><semantics id="S2.p6.13.m5.1a"><msub id="S2.p6.13.m5.1.1" xref="S2.p6.13.m5.1.1.cmml"><mi id="S2.p6.13.m5.1.1.2" xref="S2.p6.13.m5.1.1.2.cmml">k</mi><mi id="S2.p6.13.m5.1.1.3" xref="S2.p6.13.m5.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p6.13.m5.1b"><apply id="S2.p6.13.m5.1.1.cmml" xref="S2.p6.13.m5.1.1"><csymbol cd="ambiguous" id="S2.p6.13.m5.1.1.1.cmml" xref="S2.p6.13.m5.1.1">subscript</csymbol><ci id="S2.p6.13.m5.1.1.2.cmml" xref="S2.p6.13.m5.1.1.2">𝑘</ci><ci id="S2.p6.13.m5.1.1.3.cmml" xref="S2.p6.13.m5.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.13.m5.1c">k_{m}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.13.m5.1d">italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> converges to the eigenvalues of the continuous case on the interval <math alttext="[0,L]" class="ltx_Math" display="inline" id="S2.p6.14.m6.2"><semantics id="S2.p6.14.m6.2a"><mrow id="S2.p6.14.m6.2.3.2" xref="S2.p6.14.m6.2.3.1.cmml"><mo id="S2.p6.14.m6.2.3.2.1" stretchy="false" xref="S2.p6.14.m6.2.3.1.cmml">[</mo><mn id="S2.p6.14.m6.1.1" xref="S2.p6.14.m6.1.1.cmml">0</mn><mo id="S2.p6.14.m6.2.3.2.2" xref="S2.p6.14.m6.2.3.1.cmml">,</mo><mi id="S2.p6.14.m6.2.2" xref="S2.p6.14.m6.2.2.cmml">L</mi><mo id="S2.p6.14.m6.2.3.2.3" stretchy="false" xref="S2.p6.14.m6.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.14.m6.2b"><interval closure="closed" id="S2.p6.14.m6.2.3.1.cmml" xref="S2.p6.14.m6.2.3.2"><cn id="S2.p6.14.m6.1.1.cmml" type="integer" xref="S2.p6.14.m6.1.1">0</cn><ci id="S2.p6.14.m6.2.2.cmml" xref="S2.p6.14.m6.2.2">𝐿</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.14.m6.2c">[0,L]</annotation><annotation encoding="application/x-llamapun" id="S2.p6.14.m6.2d">[ 0 , italic_L ]</annotation></semantics></math>, as the following calculation shows</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx10"> <tbody id="S2.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}\lim_{a\to 0}k_{m}=\pm&\sqrt{\lim_{a\to 0}\frac{2-2% \cos\left(\frac{\pi ma}{L}\right)}{a^{2}}}\\ &=\pm\frac{\pi m}{L}\sqrt{\lim_{a\to 0}\frac{\sin\left(\frac{\pi ma}{L}\right)% }{\frac{\pi ma}{L}}}=\pm\frac{\pi m}{L}.\end{split}" class="ltx_Math" display="inline" id="S2.E13.m1.17"><semantics id="S2.E13.m1.17a"><mtable columnspacing="0pt" id="S2.E13.m1.17.17.2" rowspacing="0pt"><mtr id="S2.E13.m1.17.17.2a"><mtd class="ltx_align_right" columnalign="right" id="S2.E13.m1.17.17.2b"><mrow id="S2.E13.m1.6.6.6.6.6a"><mrow id="S2.E13.m1.6.6.6.6.6a.7"><munder id="S2.E13.m1.6.6.6.6.6a.7.1"><mo id="S2.E13.m1.1.1.1.1.1.1" movablelimits="false" xref="S2.E13.m1.1.1.1.1.1.1.cmml">lim</mo><mrow id="S2.E13.m1.2.2.2.2.2.2.1" xref="S2.E13.m1.2.2.2.2.2.2.1.cmml"><mi id="S2.E13.m1.2.2.2.2.2.2.1.2" xref="S2.E13.m1.2.2.2.2.2.2.1.2.cmml">a</mi><mo id="S2.E13.m1.2.2.2.2.2.2.1.1" stretchy="false" xref="S2.E13.m1.2.2.2.2.2.2.1.1.cmml">→</mo><mn id="S2.E13.m1.2.2.2.2.2.2.1.3" xref="S2.E13.m1.2.2.2.2.2.2.1.3.cmml">0</mn></mrow></munder><msub id="S2.E13.m1.6.6.6.6.6a.7.2"><mi id="S2.E13.m1.3.3.3.3.3.3" xref="S2.E13.m1.3.3.3.3.3.3.cmml">k</mi><mi id="S2.E13.m1.4.4.4.4.4.4.1" xref="S2.E13.m1.4.4.4.4.4.4.1.cmml">m</mi></msub></mrow><mo id="S2.E13.m1.5.5.5.5.5.5" rspace="0em" xref="S2.E13.m1.5.5.5.5.5.5.cmml">=</mo><mo id="S2.E13.m1.6.6.6.6.6.6" lspace="0em" xref="S2.E13.m1.6.6.6.6.6.6.cmml">±</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.E13.m1.17.17.2c"><msqrt id="S2.E13.m1.7.7.7.7.1.1" xref="S2.E13.m1.7.7.7.7.1.1.cmml"><mrow id="S2.E13.m1.7.7.7.7.1.1.2a" xref="S2.E13.m1.7.7.7.7.1.1.2a.cmml"><munder id="S2.E13.m1.7.7.7.7.1.1.2a.3" xref="S2.E13.m1.7.7.7.7.1.1.2a.3.cmml"><mo id="S2.E13.m1.7.7.7.7.1.1.2a.3.2" movablelimits="false" xref="S2.E13.m1.7.7.7.7.1.1.2a.3.2.cmml">lim</mo><mrow id="S2.E13.m1.7.7.7.7.1.1.2a.3.3" xref="S2.E13.m1.7.7.7.7.1.1.2a.3.3.cmml"><mi id="S2.E13.m1.7.7.7.7.1.1.2a.3.3.2" xref="S2.E13.m1.7.7.7.7.1.1.2a.3.3.2.cmml">a</mi><mo id="S2.E13.m1.7.7.7.7.1.1.2a.3.3.1" stretchy="false" xref="S2.E13.m1.7.7.7.7.1.1.2a.3.3.1.cmml">→</mo><mn id="S2.E13.m1.7.7.7.7.1.1.2a.3.3.3" xref="S2.E13.m1.7.7.7.7.1.1.2a.3.3.3.cmml">0</mn></mrow></munder><mstyle displaystyle="true" id="S2.E13.m1.7.7.7.7.1.1.2.2" xref="S2.E13.m1.7.7.7.7.1.1.2.2.cmml"><mfrac id="S2.E13.m1.7.7.7.7.1.1.2.2a" xref="S2.E13.m1.7.7.7.7.1.1.2.2.cmml"><mrow id="S2.E13.m1.7.7.7.7.1.1.2.2.2" xref="S2.E13.m1.7.7.7.7.1.1.2.2.2.cmml"><mn id="S2.E13.m1.7.7.7.7.1.1.2.2.2.4" xref="S2.E13.m1.7.7.7.7.1.1.2.2.2.4.cmml">2</mn><mo id="S2.E13.m1.7.7.7.7.1.1.2.2.2.3" xref="S2.E13.m1.7.7.7.7.1.1.2.2.2.3.cmml">−</mo><mrow id="S2.E13.m1.7.7.7.7.1.1.2.2.2.5" xref="S2.E13.m1.7.7.7.7.1.1.2.2.2.5.cmml"><mn id="S2.E13.m1.7.7.7.7.1.1.2.2.2.5.2" xref="S2.E13.m1.7.7.7.7.1.1.2.2.2.5.2.cmml">2</mn><mo id="S2.E13.m1.7.7.7.7.1.1.2.2.2.5.1" lspace="0.167em" xref="S2.E13.m1.7.7.7.7.1.1.2.2.2.5.1.cmml">⁢</mo><mrow id="S2.E13.m1.7.7.7.7.1.1.2.2.2.5.3.2" xref="S2.E13.m1.7.7.7.7.1.1.2.2.2.5.3.1.cmml"><mi id="S2.E13.m1.7.7.7.7.1.1.1.1.1.1" xref="S2.E13.m1.7.7.7.7.1.1.1.1.1.1.cmml">cos</mi><mo 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m}{L}\sqrt{\lim_{a\to 0}\frac{\sin\left(\frac{\pi ma}{L}\right)% }{\frac{\pi ma}{L}}}=\pm\frac{\pi m}{L}.\end{split}</annotation><annotation encoding="application/x-llamapun" id="S2.E13.m1.17d">start_ROW start_CELL roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ± end_CELL start_CELL square-root start_ARG roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT divide start_ARG 2 - 2 roman_cos ( divide start_ARG italic_π italic_m italic_a end_ARG start_ARG italic_L end_ARG ) end_ARG start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = ± divide start_ARG italic_π italic_m end_ARG start_ARG italic_L end_ARG square-root start_ARG roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT divide start_ARG roman_sin ( divide start_ARG italic_π italic_m italic_a end_ARG start_ARG italic_L end_ARG ) end_ARG start_ARG divide start_ARG italic_π italic_m italic_a end_ARG start_ARG italic_L end_ARG end_ARG end_ARG = ± divide start_ARG italic_π italic_m end_ARG start_ARG italic_L end_ARG . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.1">Analogously one can impose Neumann boundary condition as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Psi^{(a)}(a)-\Psi^{(a)}(0)=0,\quad\Psi^{(a)}(Na)-\Psi^{(a)}(Na-a)=0," class="ltx_Math" display="block" id="S2.E14.m1.7"><semantics id="S2.E14.m1.7a"><mrow id="S2.E14.m1.7.7.1"><mrow 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xref="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.1"></minus><apply id="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2.cmml" xref="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2"><times id="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2.1.cmml" xref="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2.1"></times><ci id="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2.2.cmml" xref="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2.2">𝑁</ci><ci id="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2.3.cmml" xref="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.2.3">𝑎</ci></apply><ci id="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.3.cmml" xref="S2.E14.m1.7.7.1.1.2.2.2.2.1.1.1.3">𝑎</ci></apply></apply></apply><cn id="S2.E14.m1.7.7.1.1.2.2.4.cmml" type="integer" xref="S2.E14.m1.7.7.1.1.2.2.4">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E14.m1.7c">\Psi^{(a)}(a)-\Psi^{(a)}(0)=0,\quad\Psi^{(a)}(Na)-\Psi^{(a)}(Na-a)=0,</annotation><annotation encoding="application/x-llamapun" id="S2.E14.m1.7d">roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_a ) - roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( 0 ) = 0 , roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_N italic_a ) - roman_Ψ start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ( italic_N italic_a - italic_a ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p7.2">which leads to the following homogeneous system of linear equations</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx11"> <tbody id="S2.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle A(g_{+}(a)-1" class="ltx_math_unparsed" display="inline" id="S2.Ex7.m1.1"><semantics id="S2.Ex7.m1.1a"><mrow id="S2.Ex7.m1.1b"><mi id="S2.Ex7.m1.1.2">A</mi><mrow id="S2.Ex7.m1.1.3"><mo id="S2.Ex7.m1.1.3.1" stretchy="false">(</mo><msub id="S2.Ex7.m1.1.3.2"><mi id="S2.Ex7.m1.1.3.2.2">g</mi><mo id="S2.Ex7.m1.1.3.2.3">+</mo></msub><mrow id="S2.Ex7.m1.1.3.3"><mo id="S2.Ex7.m1.1.3.3.1" stretchy="false">(</mo><mi id="S2.Ex7.m1.1.1">a</mi><mo id="S2.Ex7.m1.1.3.3.2" stretchy="false">)</mo></mrow><mo id="S2.Ex7.m1.1.3.4">−</mo><mn id="S2.Ex7.m1.1.3.5">1</mn></mrow></mrow><annotation encoding="application/x-tex" id="S2.Ex7.m1.1c">\displaystyle A(g_{+}(a)-1</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m1.1d">italic_A ( italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) - 1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle)+B(g_{-}(a)-1)=0," class="ltx_math_unparsed" display="inline" id="S2.Ex7.m2.1"><semantics id="S2.Ex7.m2.1a"><mrow id="S2.Ex7.m2.1b"><mo id="S2.Ex7.m2.1.1" stretchy="false">)</mo><mo id="S2.Ex7.m2.1.2">+</mo><mi id="S2.Ex7.m2.1.3">B</mi><mo id="S2.Ex7.m2.1.4" stretchy="false">(</mo><msub id="S2.Ex7.m2.1.5"><mi id="S2.Ex7.m2.1.5.2">g</mi><mo id="S2.Ex7.m2.1.5.3">−</mo></msub><mrow id="S2.Ex7.m2.1.6"><mo id="S2.Ex7.m2.1.6.1" stretchy="false">(</mo><mi id="S2.Ex7.m2.1.6.2">a</mi><mo id="S2.Ex7.m2.1.6.3" stretchy="false">)</mo></mrow><mo id="S2.Ex7.m2.1.7">−</mo><mn id="S2.Ex7.m2.1.8">1</mn><mo id="S2.Ex7.m2.1.9" stretchy="false">)</mo><mo id="S2.Ex7.m2.1.10">=</mo><mn id="S2.Ex7.m2.1.11">0</mn><mo id="S2.Ex7.m2.1.12">,</mo></mrow><annotation encoding="application/x-tex" id="S2.Ex7.m2.1c">\displaystyle)+B(g_{-}(a)-1)=0,</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m2.1d">) + italic_B ( italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) - 1 ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.E15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle A(g_{+}(a)^{N}" class="ltx_math_unparsed" display="inline" id="S2.E15.m1.1"><semantics id="S2.E15.m1.1a"><mrow id="S2.E15.m1.1b"><mi id="S2.E15.m1.1.2">A</mi><mrow id="S2.E15.m1.1.3"><mo id="S2.E15.m1.1.3.1" stretchy="false">(</mo><msub id="S2.E15.m1.1.3.2"><mi id="S2.E15.m1.1.3.2.2">g</mi><mo id="S2.E15.m1.1.3.2.3">+</mo></msub><msup id="S2.E15.m1.1.3.3"><mrow id="S2.E15.m1.1.3.3.2"><mo id="S2.E15.m1.1.3.3.2.1" stretchy="false">(</mo><mi id="S2.E15.m1.1.1">a</mi><mo id="S2.E15.m1.1.3.3.2.2" stretchy="false">)</mo></mrow><mi id="S2.E15.m1.1.3.3.3">N</mi></msup></mrow></mrow><annotation encoding="application/x-tex" id="S2.E15.m1.1c">\displaystyle A(g_{+}(a)^{N}</annotation><annotation encoding="application/x-llamapun" id="S2.E15.m1.1d">italic_A ( italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-g_{+}(a)^{N-1})" class="ltx_math_unparsed" display="inline" id="S2.E15.m2.1"><semantics id="S2.E15.m2.1a"><mrow id="S2.E15.m2.1b"><mo id="S2.E15.m2.1.2">−</mo><msub id="S2.E15.m2.1.3"><mi id="S2.E15.m2.1.3.2">g</mi><mo id="S2.E15.m2.1.3.3">+</mo></msub><msup id="S2.E15.m2.1.4"><mrow id="S2.E15.m2.1.4.2"><mo id="S2.E15.m2.1.4.2.1" stretchy="false">(</mo><mi id="S2.E15.m2.1.1">a</mi><mo id="S2.E15.m2.1.4.2.2" stretchy="false">)</mo></mrow><mrow id="S2.E15.m2.1.4.3"><mi id="S2.E15.m2.1.4.3.2">N</mi><mo id="S2.E15.m2.1.4.3.1">−</mo><mn id="S2.E15.m2.1.4.3.3">1</mn></mrow></msup><mo id="S2.E15.m2.1.5" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S2.E15.m2.1c">\displaystyle-g_{+}(a)^{N-1})</annotation><annotation encoding="application/x-llamapun" id="S2.E15.m2.1d">- italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> <tbody id="S2.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+B(g_{-}(a)^{N}-g_{-}(a)^{N-1})=0." class="ltx_Math" display="inline" id="S2.Ex8.m1.3"><semantics id="S2.Ex8.m1.3a"><mrow id="S2.Ex8.m1.3.3.1" xref="S2.Ex8.m1.3.3.1.1.cmml"><mrow id="S2.Ex8.m1.3.3.1.1" xref="S2.Ex8.m1.3.3.1.1.cmml"><mrow id="S2.Ex8.m1.3.3.1.1.1" xref="S2.Ex8.m1.3.3.1.1.1.cmml"><mo id="S2.Ex8.m1.3.3.1.1.1a" xref="S2.Ex8.m1.3.3.1.1.1.cmml">+</mo><mrow id="S2.Ex8.m1.3.3.1.1.1.1" xref="S2.Ex8.m1.3.3.1.1.1.1.cmml"><mi 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class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p7.3">The secular equation in this case is written as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\det\begin{pmatrix}g_{+}(a)-1&g_{-}(a)-1\\ g_{+}(a)^{N}-g_{+}(a)^{N-1}&g_{-}(a)^{N}-g_{-}(a)^{N-1}\end{pmatrix}=0." class="ltx_Math" display="block" id="S2.E16.m1.2"><semantics id="S2.E16.m1.2a"><mrow id="S2.E16.m1.2.2.1" xref="S2.E16.m1.2.2.1.1.cmml"><mrow id="S2.E16.m1.2.2.1.1" xref="S2.E16.m1.2.2.1.1.cmml"><mrow id="S2.E16.m1.2.2.1.1.2" xref="S2.E16.m1.2.2.1.1.2.cmml"><mo id="S2.E16.m1.2.2.1.1.2.1" movablelimits="false" rspace="0em" xref="S2.E16.m1.2.2.1.1.2.1.cmml">det</mo><mrow id="S2.E16.m1.1.1.3" xref="S2.E16.m1.1.1.2.cmml"><mo id="S2.E16.m1.1.1.3.1" xref="S2.E16.m1.1.1.2.1.cmml">(</mo><mtable 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id="S2.E16.m1.2c">\det\begin{pmatrix}g_{+}(a)-1&g_{-}(a)-1\\ g_{+}(a)^{N}-g_{+}(a)^{N-1}&g_{-}(a)^{N}-g_{-}(a)^{N-1}\end{pmatrix}=0.</annotation><annotation encoding="application/x-llamapun" id="S2.E16.m1.2d">roman_det ( start_ARG start_ROW start_CELL italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) - 1 end_CELL start_CELL italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) - 1 end_CELL end_ROW start_ROW start_CELL italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT - italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT end_CELL start_CELL italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT - italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ) = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.1">Substitution of the above definition of <math alttext="g_{\pm}(a)" class="ltx_Math" display="inline" id="S2.p8.1.m1.1"><semantics id="S2.p8.1.m1.1a"><mrow id="S2.p8.1.m1.1.2" xref="S2.p8.1.m1.1.2.cmml"><msub id="S2.p8.1.m1.1.2.2" xref="S2.p8.1.m1.1.2.2.cmml"><mi id="S2.p8.1.m1.1.2.2.2" xref="S2.p8.1.m1.1.2.2.2.cmml">g</mi><mo id="S2.p8.1.m1.1.2.2.3" xref="S2.p8.1.m1.1.2.2.3.cmml">±</mo></msub><mo id="S2.p8.1.m1.1.2.1" xref="S2.p8.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p8.1.m1.1.2.3.2" xref="S2.p8.1.m1.1.2.cmml"><mo id="S2.p8.1.m1.1.2.3.2.1" stretchy="false" xref="S2.p8.1.m1.1.2.cmml">(</mo><mi id="S2.p8.1.m1.1.1" xref="S2.p8.1.m1.1.1.cmml">a</mi><mo id="S2.p8.1.m1.1.2.3.2.2" stretchy="false" xref="S2.p8.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.1.m1.1b"><apply id="S2.p8.1.m1.1.2.cmml" xref="S2.p8.1.m1.1.2"><times id="S2.p8.1.m1.1.2.1.cmml" xref="S2.p8.1.m1.1.2.1"></times><apply id="S2.p8.1.m1.1.2.2.cmml" xref="S2.p8.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.p8.1.m1.1.2.2.1.cmml" xref="S2.p8.1.m1.1.2.2">subscript</csymbol><ci id="S2.p8.1.m1.1.2.2.2.cmml" xref="S2.p8.1.m1.1.2.2.2">𝑔</ci><csymbol cd="latexml" id="S2.p8.1.m1.1.2.2.3.cmml" xref="S2.p8.1.m1.1.2.2.3">plus-or-minus</csymbol></apply><ci id="S2.p8.1.m1.1.1.cmml" xref="S2.p8.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.1.m1.1c">g_{\pm}(a)</annotation><annotation encoding="application/x-llamapun" id="S2.p8.1.m1.1d">italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math> to the determinant equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E16" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">16</span></a>) yields to the following</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx12"> <tbody id="S2.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(e^{i\varphi_{+}(a)}-1)(e^{i\varphi_{-}(a)}-1)(e^{i\varphi_{+}(a)% (N-1)}-e^{i\varphi_{-}(a)(N-1)})=0," class="ltx_Math" display="inline" id="S2.Ex9.m1.7"><semantics id="S2.Ex9.m1.7a"><mrow id="S2.Ex9.m1.7.7.1" xref="S2.Ex9.m1.7.7.1.1.cmml"><mrow id="S2.Ex9.m1.7.7.1.1" xref="S2.Ex9.m1.7.7.1.1.cmml"><mrow id="S2.Ex9.m1.7.7.1.1.3" xref="S2.Ex9.m1.7.7.1.1.3.cmml"><mrow id="S2.Ex9.m1.7.7.1.1.1.1.1" xref="S2.Ex9.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S2.Ex9.m1.7.7.1.1.1.1.1.2" 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encoding="application/x-tex" id="S2.Ex9.m1.7c">\displaystyle(e^{i\varphi_{+}(a)}-1)(e^{i\varphi_{-}(a)}-1)(e^{i\varphi_{+}(a)% (N-1)}-e^{i\varphi_{-}(a)(N-1)})=0,</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m1.7d">( italic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) end_POSTSUPERSCRIPT - 1 ) ( italic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) end_POSTSUPERSCRIPT - 1 ) ( italic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) ( italic_N - 1 ) end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) ( italic_N - 1 ) end_POSTSUPERSCRIPT ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p8.5">or</p> <table class="ltx_equationgroup ltx_eqn_eqnarray 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id="S2.Ex10.m1.4.4.1.1.2.2.3.3.2.cmml" xref="S2.Ex10.m1.4.4.1.1.2.2.3.3.2">𝑁</ci><cn id="S2.Ex10.m1.4.4.1.1.2.2.3.3.3.cmml" type="integer" xref="S2.Ex10.m1.4.4.1.1.2.2.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex10.m1.4c">\displaystyle\varphi_{\pm}(a)=2\pi p,\quad\varphi_{+}(a)-\varphi_{-}(a)=\frac{% 2\pi m}{N-1},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex10.m1.4d">italic_φ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a ) = 2 italic_π italic_p , italic_φ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) - italic_φ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a ) = divide start_ARG 2 italic_π italic_m end_ARG start_ARG italic_N - 1 end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p8.4">where <math alttext="p" class="ltx_Math" display="inline" id="S2.p8.2.m1.1"><semantics id="S2.p8.2.m1.1a"><mi id="S2.p8.2.m1.1.1" xref="S2.p8.2.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.p8.2.m1.1b"><ci id="S2.p8.2.m1.1.1.cmml" xref="S2.p8.2.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.2.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.p8.2.m1.1d">italic_p</annotation></semantics></math> and <math alttext="m" class="ltx_Math" display="inline" id="S2.p8.3.m2.1"><semantics id="S2.p8.3.m2.1a"><mi id="S2.p8.3.m2.1.1" xref="S2.p8.3.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.p8.3.m2.1b"><ci id="S2.p8.3.m2.1.1.cmml" xref="S2.p8.3.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.3.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p8.3.m2.1d">italic_m</annotation></semantics></math> are integer numbers. Using the expression of <math alttext="\varphi_{\pm}(a)" class="ltx_Math" display="inline" id="S2.p8.4.m3.1"><semantics id="S2.p8.4.m3.1a"><mrow id="S2.p8.4.m3.1.2" xref="S2.p8.4.m3.1.2.cmml"><msub id="S2.p8.4.m3.1.2.2" xref="S2.p8.4.m3.1.2.2.cmml"><mi id="S2.p8.4.m3.1.2.2.2" xref="S2.p8.4.m3.1.2.2.2.cmml">φ</mi><mo id="S2.p8.4.m3.1.2.2.3" xref="S2.p8.4.m3.1.2.2.3.cmml">±</mo></msub><mo id="S2.p8.4.m3.1.2.1" xref="S2.p8.4.m3.1.2.1.cmml">⁢</mo><mrow id="S2.p8.4.m3.1.2.3.2" xref="S2.p8.4.m3.1.2.cmml"><mo id="S2.p8.4.m3.1.2.3.2.1" stretchy="false" xref="S2.p8.4.m3.1.2.cmml">(</mo><mi id="S2.p8.4.m3.1.1" xref="S2.p8.4.m3.1.1.cmml">a</mi><mo id="S2.p8.4.m3.1.2.3.2.2" stretchy="false" xref="S2.p8.4.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.4.m3.1b"><apply id="S2.p8.4.m3.1.2.cmml" xref="S2.p8.4.m3.1.2"><times id="S2.p8.4.m3.1.2.1.cmml" xref="S2.p8.4.m3.1.2.1"></times><apply id="S2.p8.4.m3.1.2.2.cmml" xref="S2.p8.4.m3.1.2.2"><csymbol cd="ambiguous" id="S2.p8.4.m3.1.2.2.1.cmml" xref="S2.p8.4.m3.1.2.2">subscript</csymbol><ci id="S2.p8.4.m3.1.2.2.2.cmml" xref="S2.p8.4.m3.1.2.2.2">𝜑</ci><csymbol cd="latexml" id="S2.p8.4.m3.1.2.2.3.cmml" xref="S2.p8.4.m3.1.2.2.3">plus-or-minus</csymbol></apply><ci id="S2.p8.4.m3.1.1.cmml" xref="S2.p8.4.m3.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.4.m3.1c">\varphi_{\pm}(a)</annotation><annotation encoding="application/x-llamapun" id="S2.p8.4.m3.1d">italic_φ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math>, one can derive the eigenvalues as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx14"> <tbody id="S2.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle k_{m}=\pm\frac{1}{a}\sqrt{2-2\cos\left(\frac{\pi m}{N-1}\right)}." class="ltx_Math" display="inline" id="S2.E17.m1.3"><semantics id="S2.E17.m1.3a"><mrow id="S2.E17.m1.3.3.1" xref="S2.E17.m1.3.3.1.1.cmml"><mrow id="S2.E17.m1.3.3.1.1" xref="S2.E17.m1.3.3.1.1.cmml"><msub id="S2.E17.m1.3.3.1.1.2" xref="S2.E17.m1.3.3.1.1.2.cmml"><mi id="S2.E17.m1.3.3.1.1.2.2" xref="S2.E17.m1.3.3.1.1.2.2.cmml">k</mi><mi id="S2.E17.m1.3.3.1.1.2.3" xref="S2.E17.m1.3.3.1.1.2.3.cmml">m</mi></msub><mo id="S2.E17.m1.3.3.1.1.1" xref="S2.E17.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E17.m1.3.3.1.1.3" xref="S2.E17.m1.3.3.1.1.3.cmml"><mo id="S2.E17.m1.3.3.1.1.3a" xref="S2.E17.m1.3.3.1.1.3.cmml">±</mo><mrow id="S2.E17.m1.3.3.1.1.3.2" xref="S2.E17.m1.3.3.1.1.3.2.cmml"><mstyle displaystyle="true" id="S2.E17.m1.3.3.1.1.3.2.2" xref="S2.E17.m1.3.3.1.1.3.2.2.cmml"><mfrac id="S2.E17.m1.3.3.1.1.3.2.2a" xref="S2.E17.m1.3.3.1.1.3.2.2.cmml"><mn id="S2.E17.m1.3.3.1.1.3.2.2.2" xref="S2.E17.m1.3.3.1.1.3.2.2.2.cmml">1</mn><mi id="S2.E17.m1.3.3.1.1.3.2.2.3" xref="S2.E17.m1.3.3.1.1.3.2.2.3.cmml">a</mi></mfrac></mstyle><mo id="S2.E17.m1.3.3.1.1.3.2.1" xref="S2.E17.m1.3.3.1.1.3.2.1.cmml">⁢</mo><msqrt id="S2.E17.m1.2.2" xref="S2.E17.m1.2.2.cmml"><mrow id="S2.E17.m1.2.2.2" xref="S2.E17.m1.2.2.2.cmml"><mn id="S2.E17.m1.2.2.2.4" xref="S2.E17.m1.2.2.2.4.cmml">2</mn><mo id="S2.E17.m1.2.2.2.3" xref="S2.E17.m1.2.2.2.3.cmml">−</mo><mrow id="S2.E17.m1.2.2.2.5" xref="S2.E17.m1.2.2.2.5.cmml"><mn id="S2.E17.m1.2.2.2.5.2" xref="S2.E17.m1.2.2.2.5.2.cmml">2</mn><mo id="S2.E17.m1.2.2.2.5.1" lspace="0.167em" xref="S2.E17.m1.2.2.2.5.1.cmml">⁢</mo><mrow id="S2.E17.m1.2.2.2.5.3.2" xref="S2.E17.m1.2.2.2.5.3.1.cmml"><mi id="S2.E17.m1.1.1.1.1" xref="S2.E17.m1.1.1.1.1.cmml">cos</mi><mo id="S2.E17.m1.2.2.2.5.3.2a" xref="S2.E17.m1.2.2.2.5.3.1.cmml">⁡</mo><mrow id="S2.E17.m1.2.2.2.5.3.2.1" xref="S2.E17.m1.2.2.2.5.3.1.cmml"><mo id="S2.E17.m1.2.2.2.5.3.2.1.1" xref="S2.E17.m1.2.2.2.5.3.1.cmml">(</mo><mstyle displaystyle="true" id="S2.E17.m1.2.2.2.2" xref="S2.E17.m1.2.2.2.2.cmml"><mfrac id="S2.E17.m1.2.2.2.2a" xref="S2.E17.m1.2.2.2.2.cmml"><mrow id="S2.E17.m1.2.2.2.2.2" xref="S2.E17.m1.2.2.2.2.2.cmml"><mi id="S2.E17.m1.2.2.2.2.2.2" xref="S2.E17.m1.2.2.2.2.2.2.cmml">π</mi><mo id="S2.E17.m1.2.2.2.2.2.1" xref="S2.E17.m1.2.2.2.2.2.1.cmml">⁢</mo><mi id="S2.E17.m1.2.2.2.2.2.3" xref="S2.E17.m1.2.2.2.2.2.3.cmml">m</mi></mrow><mrow id="S2.E17.m1.2.2.2.2.3" xref="S2.E17.m1.2.2.2.2.3.cmml"><mi id="S2.E17.m1.2.2.2.2.3.2" xref="S2.E17.m1.2.2.2.2.3.2.cmml">N</mi><mo id="S2.E17.m1.2.2.2.2.3.1" xref="S2.E17.m1.2.2.2.2.3.1.cmml">−</mo><mn id="S2.E17.m1.2.2.2.2.3.3" xref="S2.E17.m1.2.2.2.2.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S2.E17.m1.2.2.2.5.3.2.1.2" xref="S2.E17.m1.2.2.2.5.3.1.cmml">)</mo></mrow></mrow></mrow></mrow></msqrt></mrow></mrow></mrow><mo id="S2.E17.m1.3.3.1.2" lspace="0em" xref="S2.E17.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E17.m1.3b"><apply id="S2.E17.m1.3.3.1.1.cmml" xref="S2.E17.m1.3.3.1"><eq id="S2.E17.m1.3.3.1.1.1.cmml" xref="S2.E17.m1.3.3.1.1.1"></eq><apply id="S2.E17.m1.3.3.1.1.2.cmml" xref="S2.E17.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.E17.m1.3.3.1.1.2.1.cmml" xref="S2.E17.m1.3.3.1.1.2">subscript</csymbol><ci id="S2.E17.m1.3.3.1.1.2.2.cmml" xref="S2.E17.m1.3.3.1.1.2.2">𝑘</ci><ci id="S2.E17.m1.3.3.1.1.2.3.cmml" xref="S2.E17.m1.3.3.1.1.2.3">𝑚</ci></apply><apply id="S2.E17.m1.3.3.1.1.3.cmml" xref="S2.E17.m1.3.3.1.1.3"><csymbol cd="latexml" id="S2.E17.m1.3.3.1.1.3.1.cmml" xref="S2.E17.m1.3.3.1.1.3">plus-or-minus</csymbol><apply id="S2.E17.m1.3.3.1.1.3.2.cmml" xref="S2.E17.m1.3.3.1.1.3.2"><times id="S2.E17.m1.3.3.1.1.3.2.1.cmml" xref="S2.E17.m1.3.3.1.1.3.2.1"></times><apply id="S2.E17.m1.3.3.1.1.3.2.2.cmml" xref="S2.E17.m1.3.3.1.1.3.2.2"><divide id="S2.E17.m1.3.3.1.1.3.2.2.1.cmml" xref="S2.E17.m1.3.3.1.1.3.2.2"></divide><cn id="S2.E17.m1.3.3.1.1.3.2.2.2.cmml" type="integer" xref="S2.E17.m1.3.3.1.1.3.2.2.2">1</cn><ci id="S2.E17.m1.3.3.1.1.3.2.2.3.cmml" xref="S2.E17.m1.3.3.1.1.3.2.2.3">𝑎</ci></apply><apply id="S2.E17.m1.2.2.cmml" xref="S2.E17.m1.2.2"><root id="S2.E17.m1.2.2a.cmml" xref="S2.E17.m1.2.2"></root><apply id="S2.E17.m1.2.2.2.cmml" xref="S2.E17.m1.2.2.2"><minus id="S2.E17.m1.2.2.2.3.cmml" xref="S2.E17.m1.2.2.2.3"></minus><cn id="S2.E17.m1.2.2.2.4.cmml" type="integer" xref="S2.E17.m1.2.2.2.4">2</cn><apply id="S2.E17.m1.2.2.2.5.cmml" xref="S2.E17.m1.2.2.2.5"><times id="S2.E17.m1.2.2.2.5.1.cmml" xref="S2.E17.m1.2.2.2.5.1"></times><cn id="S2.E17.m1.2.2.2.5.2.cmml" type="integer" xref="S2.E17.m1.2.2.2.5.2">2</cn><apply id="S2.E17.m1.2.2.2.5.3.1.cmml" xref="S2.E17.m1.2.2.2.5.3.2"><cos id="S2.E17.m1.1.1.1.1.cmml" xref="S2.E17.m1.1.1.1.1"></cos><apply id="S2.E17.m1.2.2.2.2.cmml" xref="S2.E17.m1.2.2.2.2"><divide id="S2.E17.m1.2.2.2.2.1.cmml" xref="S2.E17.m1.2.2.2.2"></divide><apply id="S2.E17.m1.2.2.2.2.2.cmml" xref="S2.E17.m1.2.2.2.2.2"><times id="S2.E17.m1.2.2.2.2.2.1.cmml" xref="S2.E17.m1.2.2.2.2.2.1"></times><ci id="S2.E17.m1.2.2.2.2.2.2.cmml" xref="S2.E17.m1.2.2.2.2.2.2">𝜋</ci><ci id="S2.E17.m1.2.2.2.2.2.3.cmml" xref="S2.E17.m1.2.2.2.2.2.3">𝑚</ci></apply><apply id="S2.E17.m1.2.2.2.2.3.cmml" xref="S2.E17.m1.2.2.2.2.3"><minus id="S2.E17.m1.2.2.2.2.3.1.cmml" xref="S2.E17.m1.2.2.2.2.3.1"></minus><ci id="S2.E17.m1.2.2.2.2.3.2.cmml" xref="S2.E17.m1.2.2.2.2.3.2">𝑁</ci><cn id="S2.E17.m1.2.2.2.2.3.3.cmml" type="integer" xref="S2.E17.m1.2.2.2.2.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E17.m1.3c">\displaystyle k_{m}=\pm\frac{1}{a}\sqrt{2-2\cos\left(\frac{\pi m}{N-1}\right)}.</annotation><annotation encoding="application/x-llamapun" id="S2.E17.m1.3d">italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ± divide start_ARG 1 end_ARG start_ARG italic_a end_ARG square-root start_ARG 2 - 2 roman_cos ( divide start_ARG italic_π italic_m end_ARG start_ARG italic_N - 1 end_ARG ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p8.6">Similarly, it is easy to show that the convergence of the eigenvalues to the continuous case as shown for the Dirichlet boundary condition as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim_{a\to 0}k_{m}=\pm\frac{\pi m}{L}." class="ltx_Math" display="block" id="S2.E18.m1.1"><semantics id="S2.E18.m1.1a"><mrow id="S2.E18.m1.1.1.1" xref="S2.E18.m1.1.1.1.1.cmml"><mrow id="S2.E18.m1.1.1.1.1" xref="S2.E18.m1.1.1.1.1.cmml"><mrow id="S2.E18.m1.1.1.1.1.2" xref="S2.E18.m1.1.1.1.1.2.cmml"><munder id="S2.E18.m1.1.1.1.1.2.1" xref="S2.E18.m1.1.1.1.1.2.1.cmml"><mo id="S2.E18.m1.1.1.1.1.2.1.2" movablelimits="false" xref="S2.E18.m1.1.1.1.1.2.1.2.cmml">lim</mo><mrow id="S2.E18.m1.1.1.1.1.2.1.3" xref="S2.E18.m1.1.1.1.1.2.1.3.cmml"><mi id="S2.E18.m1.1.1.1.1.2.1.3.2" xref="S2.E18.m1.1.1.1.1.2.1.3.2.cmml">a</mi><mo id="S2.E18.m1.1.1.1.1.2.1.3.1" stretchy="false" xref="S2.E18.m1.1.1.1.1.2.1.3.1.cmml">→</mo><mn id="S2.E18.m1.1.1.1.1.2.1.3.3" xref="S2.E18.m1.1.1.1.1.2.1.3.3.cmml">0</mn></mrow></munder><msub id="S2.E18.m1.1.1.1.1.2.2" xref="S2.E18.m1.1.1.1.1.2.2.cmml"><mi id="S2.E18.m1.1.1.1.1.2.2.2" xref="S2.E18.m1.1.1.1.1.2.2.2.cmml">k</mi><mi id="S2.E18.m1.1.1.1.1.2.2.3" xref="S2.E18.m1.1.1.1.1.2.2.3.cmml">m</mi></msub></mrow><mo id="S2.E18.m1.1.1.1.1.1" xref="S2.E18.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E18.m1.1.1.1.1.3" xref="S2.E18.m1.1.1.1.1.3.cmml"><mo id="S2.E18.m1.1.1.1.1.3a" xref="S2.E18.m1.1.1.1.1.3.cmml">±</mo><mfrac id="S2.E18.m1.1.1.1.1.3.2" xref="S2.E18.m1.1.1.1.1.3.2.cmml"><mrow id="S2.E18.m1.1.1.1.1.3.2.2" xref="S2.E18.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.E18.m1.1.1.1.1.3.2.2.2" xref="S2.E18.m1.1.1.1.1.3.2.2.2.cmml">π</mi><mo id="S2.E18.m1.1.1.1.1.3.2.2.1" xref="S2.E18.m1.1.1.1.1.3.2.2.1.cmml">⁢</mo><mi id="S2.E18.m1.1.1.1.1.3.2.2.3" xref="S2.E18.m1.1.1.1.1.3.2.2.3.cmml">m</mi></mrow><mi id="S2.E18.m1.1.1.1.1.3.2.3" xref="S2.E18.m1.1.1.1.1.3.2.3.cmml">L</mi></mfrac></mrow></mrow><mo id="S2.E18.m1.1.1.1.2" lspace="0em" xref="S2.E18.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E18.m1.1b"><apply id="S2.E18.m1.1.1.1.1.cmml" xref="S2.E18.m1.1.1.1"><eq id="S2.E18.m1.1.1.1.1.1.cmml" xref="S2.E18.m1.1.1.1.1.1"></eq><apply id="S2.E18.m1.1.1.1.1.2.cmml" xref="S2.E18.m1.1.1.1.1.2"><apply id="S2.E18.m1.1.1.1.1.2.1.cmml" xref="S2.E18.m1.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S2.E18.m1.1.1.1.1.2.1.1.cmml" xref="S2.E18.m1.1.1.1.1.2.1">subscript</csymbol><limit id="S2.E18.m1.1.1.1.1.2.1.2.cmml" 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xref="S2.E18.m1.1.1.1.1.3.2"></divide><apply id="S2.E18.m1.1.1.1.1.3.2.2.cmml" xref="S2.E18.m1.1.1.1.1.3.2.2"><times id="S2.E18.m1.1.1.1.1.3.2.2.1.cmml" xref="S2.E18.m1.1.1.1.1.3.2.2.1"></times><ci id="S2.E18.m1.1.1.1.1.3.2.2.2.cmml" xref="S2.E18.m1.1.1.1.1.3.2.2.2">𝜋</ci><ci id="S2.E18.m1.1.1.1.1.3.2.2.3.cmml" xref="S2.E18.m1.1.1.1.1.3.2.2.3">𝑚</ci></apply><ci id="S2.E18.m1.1.1.1.1.3.2.3.cmml" xref="S2.E18.m1.1.1.1.1.3.2.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E18.m1.1c">\lim_{a\to 0}k_{m}=\pm\frac{\pi m}{L}.</annotation><annotation encoding="application/x-llamapun" id="S2.E18.m1.1d">roman_lim start_POSTSUBSCRIPT italic_a → 0 end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ± divide start_ARG italic_π italic_m end_ARG start_ARG italic_L end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p9"> <p class="ltx_p" id="S2.p9.5">We collect the first five nonzero eigenvalues of continuous Schrödinger equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E1" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">1</span></a>) with Dirichlet and Neumann boundary conditions defined on the interval <math alttext="[0,1]" class="ltx_Math" display="inline" id="S2.p9.1.m1.2"><semantics id="S2.p9.1.m1.2a"><mrow id="S2.p9.1.m1.2.3.2" xref="S2.p9.1.m1.2.3.1.cmml"><mo id="S2.p9.1.m1.2.3.2.1" stretchy="false" xref="S2.p9.1.m1.2.3.1.cmml">[</mo><mn id="S2.p9.1.m1.1.1" xref="S2.p9.1.m1.1.1.cmml">0</mn><mo id="S2.p9.1.m1.2.3.2.2" xref="S2.p9.1.m1.2.3.1.cmml">,</mo><mn id="S2.p9.1.m1.2.2" xref="S2.p9.1.m1.2.2.cmml">1</mn><mo id="S2.p9.1.m1.2.3.2.3" stretchy="false" xref="S2.p9.1.m1.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p9.1.m1.2b"><interval closure="closed" id="S2.p9.1.m1.2.3.1.cmml" xref="S2.p9.1.m1.2.3.2"><cn id="S2.p9.1.m1.1.1.cmml" type="integer" xref="S2.p9.1.m1.1.1">0</cn><cn id="S2.p9.1.m1.2.2.cmml" type="integer" xref="S2.p9.1.m1.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.1.m1.2c">[0,1]</annotation><annotation encoding="application/x-llamapun" id="S2.p9.1.m1.2d">[ 0 , 1 ]</annotation></semantics></math> which are given by <math alttext="k_{m}=\pi m" class="ltx_Math" display="inline" id="S2.p9.2.m2.1"><semantics id="S2.p9.2.m2.1a"><mrow id="S2.p9.2.m2.1.1" xref="S2.p9.2.m2.1.1.cmml"><msub id="S2.p9.2.m2.1.1.2" xref="S2.p9.2.m2.1.1.2.cmml"><mi id="S2.p9.2.m2.1.1.2.2" xref="S2.p9.2.m2.1.1.2.2.cmml">k</mi><mi id="S2.p9.2.m2.1.1.2.3" xref="S2.p9.2.m2.1.1.2.3.cmml">m</mi></msub><mo id="S2.p9.2.m2.1.1.1" xref="S2.p9.2.m2.1.1.1.cmml">=</mo><mrow id="S2.p9.2.m2.1.1.3" xref="S2.p9.2.m2.1.1.3.cmml"><mi id="S2.p9.2.m2.1.1.3.2" xref="S2.p9.2.m2.1.1.3.2.cmml">π</mi><mo id="S2.p9.2.m2.1.1.3.1" xref="S2.p9.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.p9.2.m2.1.1.3.3" xref="S2.p9.2.m2.1.1.3.3.cmml">m</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p9.2.m2.1b"><apply id="S2.p9.2.m2.1.1.cmml" xref="S2.p9.2.m2.1.1"><eq id="S2.p9.2.m2.1.1.1.cmml" xref="S2.p9.2.m2.1.1.1"></eq><apply id="S2.p9.2.m2.1.1.2.cmml" xref="S2.p9.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.p9.2.m2.1.1.2.1.cmml" xref="S2.p9.2.m2.1.1.2">subscript</csymbol><ci id="S2.p9.2.m2.1.1.2.2.cmml" xref="S2.p9.2.m2.1.1.2.2">𝑘</ci><ci id="S2.p9.2.m2.1.1.2.3.cmml" xref="S2.p9.2.m2.1.1.2.3">𝑚</ci></apply><apply id="S2.p9.2.m2.1.1.3.cmml" xref="S2.p9.2.m2.1.1.3"><times id="S2.p9.2.m2.1.1.3.1.cmml" xref="S2.p9.2.m2.1.1.3.1"></times><ci id="S2.p9.2.m2.1.1.3.2.cmml" xref="S2.p9.2.m2.1.1.3.2">𝜋</ci><ci id="S2.p9.2.m2.1.1.3.3.cmml" xref="S2.p9.2.m2.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.2.m2.1c">k_{m}=\pi m</annotation><annotation encoding="application/x-llamapun" id="S2.p9.2.m2.1d">italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_π italic_m</annotation></semantics></math>, <math alttext="m=1,...,5" class="ltx_Math" display="inline" id="S2.p9.3.m3.3"><semantics id="S2.p9.3.m3.3a"><mrow id="S2.p9.3.m3.3.4" xref="S2.p9.3.m3.3.4.cmml"><mi id="S2.p9.3.m3.3.4.2" xref="S2.p9.3.m3.3.4.2.cmml">m</mi><mo id="S2.p9.3.m3.3.4.1" xref="S2.p9.3.m3.3.4.1.cmml">=</mo><mrow id="S2.p9.3.m3.3.4.3.2" xref="S2.p9.3.m3.3.4.3.1.cmml"><mn id="S2.p9.3.m3.1.1" xref="S2.p9.3.m3.1.1.cmml">1</mn><mo id="S2.p9.3.m3.3.4.3.2.1" xref="S2.p9.3.m3.3.4.3.1.cmml">,</mo><mi id="S2.p9.3.m3.2.2" mathvariant="normal" xref="S2.p9.3.m3.2.2.cmml">…</mi><mo id="S2.p9.3.m3.3.4.3.2.2" xref="S2.p9.3.m3.3.4.3.1.cmml">,</mo><mn id="S2.p9.3.m3.3.3" xref="S2.p9.3.m3.3.3.cmml">5</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p9.3.m3.3b"><apply id="S2.p9.3.m3.3.4.cmml" xref="S2.p9.3.m3.3.4"><eq id="S2.p9.3.m3.3.4.1.cmml" xref="S2.p9.3.m3.3.4.1"></eq><ci id="S2.p9.3.m3.3.4.2.cmml" xref="S2.p9.3.m3.3.4.2">𝑚</ci><list id="S2.p9.3.m3.3.4.3.1.cmml" xref="S2.p9.3.m3.3.4.3.2"><cn id="S2.p9.3.m3.1.1.cmml" type="integer" xref="S2.p9.3.m3.1.1">1</cn><ci id="S2.p9.3.m3.2.2.cmml" xref="S2.p9.3.m3.2.2">…</ci><cn id="S2.p9.3.m3.3.3.cmml" type="integer" xref="S2.p9.3.m3.3.3">5</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.3.m3.3c">m=1,...,5</annotation><annotation encoding="application/x-llamapun" id="S2.p9.3.m3.3d">italic_m = 1 , … , 5</annotation></semantics></math> and compare them in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.T1" title="Table 1 ‣ 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">1</span></a> with the first five nonzero eigenvalues of the discrete Schrödinger equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E2" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a>) with Dirichlet (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E6" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">6</span></a>) and Neumann (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E14" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">14</span></a>) boundary conditions on the interval <math alttext="[0,1]" class="ltx_Math" display="inline" id="S2.p9.4.m4.2"><semantics id="S2.p9.4.m4.2a"><mrow id="S2.p9.4.m4.2.3.2" xref="S2.p9.4.m4.2.3.1.cmml"><mo id="S2.p9.4.m4.2.3.2.1" stretchy="false" xref="S2.p9.4.m4.2.3.1.cmml">[</mo><mn id="S2.p9.4.m4.1.1" xref="S2.p9.4.m4.1.1.cmml">0</mn><mo id="S2.p9.4.m4.2.3.2.2" xref="S2.p9.4.m4.2.3.1.cmml">,</mo><mn id="S2.p9.4.m4.2.2" xref="S2.p9.4.m4.2.2.cmml">1</mn><mo id="S2.p9.4.m4.2.3.2.3" stretchy="false" xref="S2.p9.4.m4.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p9.4.m4.2b"><interval closure="closed" id="S2.p9.4.m4.2.3.1.cmml" xref="S2.p9.4.m4.2.3.2"><cn id="S2.p9.4.m4.1.1.cmml" type="integer" xref="S2.p9.4.m4.1.1">0</cn><cn id="S2.p9.4.m4.2.2.cmml" type="integer" xref="S2.p9.4.m4.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.4.m4.2c">[0,1]</annotation><annotation encoding="application/x-llamapun" id="S2.p9.4.m4.2d">[ 0 , 1 ]</annotation></semantics></math>, respectively, for different values of step size <math alttext="a" class="ltx_Math" display="inline" id="S2.p9.5.m5.1"><semantics id="S2.p9.5.m5.1a"><mi id="S2.p9.5.m5.1.1" xref="S2.p9.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.p9.5.m5.1b"><ci id="S2.p9.5.m5.1.1.cmml" xref="S2.p9.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.5.m5.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.p9.5.m5.1d">italic_a</annotation></semantics></math>.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Basic theory of quantum graphs</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.8">In this section, we will briefly recall the introduction of the quantum graphs and their spectrum by following the Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib6" title="">6</a>]</cite>. A graph consists of <math alttext="V" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_V</annotation></semantics></math> vertices, where <math alttext="V" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mi id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><ci id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_V</annotation></semantics></math> is a natural number, and it is connected by <math alttext="E" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mi id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">E</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">italic_E</annotation></semantics></math> edges, <math alttext="E\in\mathbb{N}" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mrow id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mi id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml">E</mi><mo id="S3.p1.4.m4.1.1.1" xref="S3.p1.4.m4.1.1.1.cmml">∈</mo><mi id="S3.p1.4.m4.1.1.3" xref="S3.p1.4.m4.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><apply id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1"><in id="S3.p1.4.m4.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1"></in><ci id="S3.p1.4.m4.1.1.2.cmml" xref="S3.p1.4.m4.1.1.2">𝐸</ci><ci id="S3.p1.4.m4.1.1.3.cmml" xref="S3.p1.4.m4.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">E\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">italic_E ∈ blackboard_N</annotation></semantics></math>. If vertices <math alttext="i" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><mi id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><ci id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">italic_i</annotation></semantics></math> and <math alttext="j" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><mi id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><ci id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">italic_j</annotation></semantics></math> are connected we call it the <math alttext="(i,j)" class="ltx_Math" display="inline" id="S3.p1.7.m7.2"><semantics id="S3.p1.7.m7.2a"><mrow id="S3.p1.7.m7.2.3.2" xref="S3.p1.7.m7.2.3.1.cmml"><mo id="S3.p1.7.m7.2.3.2.1" stretchy="false" xref="S3.p1.7.m7.2.3.1.cmml">(</mo><mi id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml">i</mi><mo id="S3.p1.7.m7.2.3.2.2" xref="S3.p1.7.m7.2.3.1.cmml">,</mo><mi id="S3.p1.7.m7.2.2" xref="S3.p1.7.m7.2.2.cmml">j</mi><mo id="S3.p1.7.m7.2.3.2.3" stretchy="false" xref="S3.p1.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.2b"><interval closure="open" id="S3.p1.7.m7.2.3.1.cmml" xref="S3.p1.7.m7.2.3.2"><ci id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1">𝑖</ci><ci id="S3.p1.7.m7.2.2.cmml" xref="S3.p1.7.m7.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.2d">( italic_i , italic_j )</annotation></semantics></math> edge. Here we always assume that at most one edge connects the same two vertices and moreover, that there are no loops, i.e. there is no edge of the form <math alttext="(i,i)" class="ltx_Math" display="inline" id="S3.p1.8.m8.2"><semantics id="S3.p1.8.m8.2a"><mrow id="S3.p1.8.m8.2.3.2" xref="S3.p1.8.m8.2.3.1.cmml"><mo id="S3.p1.8.m8.2.3.2.1" stretchy="false" xref="S3.p1.8.m8.2.3.1.cmml">(</mo><mi id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml">i</mi><mo id="S3.p1.8.m8.2.3.2.2" xref="S3.p1.8.m8.2.3.1.cmml">,</mo><mi id="S3.p1.8.m8.2.2" xref="S3.p1.8.m8.2.2.cmml">i</mi><mo id="S3.p1.8.m8.2.3.2.3" stretchy="false" xref="S3.p1.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.2b"><interval closure="open" id="S3.p1.8.m8.2.3.1.cmml" xref="S3.p1.8.m8.2.3.2"><ci id="S3.p1.8.m8.1.1.cmml" xref="S3.p1.8.m8.1.1">𝑖</ci><ci id="S3.p1.8.m8.2.2.cmml" xref="S3.p1.8.m8.2.2">𝑖</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m8.2c">(i,i)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m8.2d">( italic_i , italic_i )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">The topology of graph is defined by its <math alttext="V\times V" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mrow id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">V</mi><mo id="S3.p2.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.p2.1.m1.1.1.1.cmml">×</mo><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><times id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1"></times><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">𝑉</ci><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">V\times V</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_V × italic_V</annotation></semantics></math> adjacency matrix as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx15"> <tbody id="S3.E19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C_{i,j}=C_{j,i}=\left\{\begin{matrix}1,&\text{if}\;i\;\text{and}% \;j\;\text{are}\;\text{connected,}\\ 0,&\text{otherwise,}\end{matrix}\right." class="ltx_math_unparsed" display="inline" id="S3.E19.m1.5"><semantics id="S3.E19.m1.5a"><mrow id="S3.E19.m1.5b"><msub id="S3.E19.m1.5.6"><mi id="S3.E19.m1.5.6.2">C</mi><mrow id="S3.E19.m1.3.3.2.4"><mi id="S3.E19.m1.2.2.1.1">i</mi><mo id="S3.E19.m1.3.3.2.4.1">,</mo><mi id="S3.E19.m1.3.3.2.2">j</mi></mrow></msub><mo id="S3.E19.m1.5.7">=</mo><msub id="S3.E19.m1.5.8"><mi id="S3.E19.m1.5.8.2">C</mi><mrow id="S3.E19.m1.5.5.2.4"><mi id="S3.E19.m1.4.4.1.1">j</mi><mo id="S3.E19.m1.5.5.2.4.1">,</mo><mi id="S3.E19.m1.5.5.2.2">i</mi></mrow></msub><mo id="S3.E19.m1.5.9">=</mo><mrow id="S3.E19.m1.5.10"><mo id="S3.E19.m1.5.10.1">{</mo><mtable columnspacing="5pt" id="S3.E19.m1.1.1.1.1" rowspacing="0pt"><mtr id="S3.E19.m1.1.1.1.1a"><mtd id="S3.E19.m1.1.1.1.1b"><mrow id="S3.E19.m1.1.1.1.1.1.1.1.1.3"><mn id="S3.E19.m1.1.1.1.1.1.1.1.1.1">1</mn><mo id="S3.E19.m1.1.1.1.1.1.1.1.1.3.1">,</mo></mrow></mtd><mtd id="S3.E19.m1.1.1.1.1c"><mrow id="S3.E19.m1.1.1.1.1.1.1.2.1"><mtext id="S3.E19.m1.1.1.1.1.1.1.2.1.2">if</mtext><mo id="S3.E19.m1.1.1.1.1.1.1.2.1.1" lspace="0.280em">⁢</mo><mi id="S3.E19.m1.1.1.1.1.1.1.2.1.3">i</mi><mo id="S3.E19.m1.1.1.1.1.1.1.2.1.1a" lspace="0.280em">⁢</mo><mtext id="S3.E19.m1.1.1.1.1.1.1.2.1.4">and</mtext><mo id="S3.E19.m1.1.1.1.1.1.1.2.1.1b" lspace="0.280em">⁢</mo><mi id="S3.E19.m1.1.1.1.1.1.1.2.1.5">j</mi><mo id="S3.E19.m1.1.1.1.1.1.1.2.1.1c" lspace="0.280em">⁢</mo><mtext id="S3.E19.m1.1.1.1.1.1.1.2.1.6">are</mtext><mo id="S3.E19.m1.1.1.1.1.1.1.2.1.1d" lspace="0.280em">⁢</mo><mtext id="S3.E19.m1.1.1.1.1.1.1.2.1.7">connected,</mtext></mrow></mtd></mtr><mtr id="S3.E19.m1.1.1.1.1d"><mtd id="S3.E19.m1.1.1.1.1e"><mrow id="S3.E19.m1.1.1.1.1.2.2.1.1.3"><mn id="S3.E19.m1.1.1.1.1.2.2.1.1.1">0</mn><mo id="S3.E19.m1.1.1.1.1.2.2.1.1.3.1">,</mo></mrow></mtd><mtd id="S3.E19.m1.1.1.1.1f"><mtext id="S3.E19.m1.1.1.1.1.2.2.2.1">otherwise,</mtext></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex" id="S3.E19.m1.5c">\displaystyle C_{i,j}=C_{j,i}=\left\{\begin{matrix}1,&\text{if}\;i\;\text{and}% \;j\;\text{are}\;\text{connected,}\\ 0,&\text{otherwise,}\end{matrix}\right.</annotation><annotation encoding="application/x-llamapun" id="S3.E19.m1.5d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT = { start_ARG start_ROW start_CELL 1 , end_CELL start_CELL if italic_i and italic_j are connected, end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise, end_CELL end_ROW end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.3">where <math alttext="i,j=1,2,3,..,V" class="ltx_math_unparsed" display="inline" id="S3.p2.2.m1.4"><semantics id="S3.p2.2.m1.4a"><mrow id="S3.p2.2.m1.4b"><mi id="S3.p2.2.m1.1.1">i</mi><mo id="S3.p2.2.m1.4.5">,</mo><mi id="S3.p2.2.m1.2.2">j</mi><mo id="S3.p2.2.m1.4.6">=</mo><mn id="S3.p2.2.m1.4.7">1</mn><mo id="S3.p2.2.m1.4.8">,</mo><mn id="S3.p2.2.m1.3.3">2</mn><mo id="S3.p2.2.m1.4.9">,</mo><mn id="S3.p2.2.m1.4.4">3</mn><mo id="S3.p2.2.m1.4.10">,</mo><mo id="S3.p2.2.m1.4.11" lspace="0em" rspace="0.0835em">.</mo><mo id="S3.p2.2.m1.4.12" lspace="0.0835em" rspace="0.167em">.</mo><mo id="S3.p2.2.m1.4.13">,</mo><mi id="S3.p2.2.m1.4.14">V</mi></mrow><annotation encoding="application/x-tex" id="S3.p2.2.m1.4c">i,j=1,2,3,..,V</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m1.4d">italic_i , italic_j = 1 , 2 , 3 , . . , italic_V</annotation></semantics></math>. The number of edges can be found in terms of the adjacency matrix by <math alttext="E=\frac{1}{2}\sum_{i,j=1}^{V}C_{i,j}" class="ltx_Math" display="inline" id="S3.p2.3.m2.4"><semantics id="S3.p2.3.m2.4a"><mrow id="S3.p2.3.m2.4.5" xref="S3.p2.3.m2.4.5.cmml"><mi id="S3.p2.3.m2.4.5.2" xref="S3.p2.3.m2.4.5.2.cmml">E</mi><mo id="S3.p2.3.m2.4.5.1" xref="S3.p2.3.m2.4.5.1.cmml">=</mo><mrow id="S3.p2.3.m2.4.5.3" xref="S3.p2.3.m2.4.5.3.cmml"><mfrac id="S3.p2.3.m2.4.5.3.2" xref="S3.p2.3.m2.4.5.3.2.cmml"><mn id="S3.p2.3.m2.4.5.3.2.2" xref="S3.p2.3.m2.4.5.3.2.2.cmml">1</mn><mn id="S3.p2.3.m2.4.5.3.2.3" xref="S3.p2.3.m2.4.5.3.2.3.cmml">2</mn></mfrac><mo id="S3.p2.3.m2.4.5.3.1" xref="S3.p2.3.m2.4.5.3.1.cmml">⁢</mo><mrow id="S3.p2.3.m2.4.5.3.3" xref="S3.p2.3.m2.4.5.3.3.cmml"><msubsup id="S3.p2.3.m2.4.5.3.3.1" xref="S3.p2.3.m2.4.5.3.3.1.cmml"><mo id="S3.p2.3.m2.4.5.3.3.1.2.2" xref="S3.p2.3.m2.4.5.3.3.1.2.2.cmml">∑</mo><mrow id="S3.p2.3.m2.2.2.2" xref="S3.p2.3.m2.2.2.2.cmml"><mrow id="S3.p2.3.m2.2.2.2.4.2" xref="S3.p2.3.m2.2.2.2.4.1.cmml"><mi id="S3.p2.3.m2.1.1.1.1" xref="S3.p2.3.m2.1.1.1.1.cmml">i</mi><mo id="S3.p2.3.m2.2.2.2.4.2.1" xref="S3.p2.3.m2.2.2.2.4.1.cmml">,</mo><mi id="S3.p2.3.m2.2.2.2.2" xref="S3.p2.3.m2.2.2.2.2.cmml">j</mi></mrow><mo id="S3.p2.3.m2.2.2.2.3" xref="S3.p2.3.m2.2.2.2.3.cmml">=</mo><mn id="S3.p2.3.m2.2.2.2.5" xref="S3.p2.3.m2.2.2.2.5.cmml">1</mn></mrow><mi id="S3.p2.3.m2.4.5.3.3.1.3" xref="S3.p2.3.m2.4.5.3.3.1.3.cmml">V</mi></msubsup><msub id="S3.p2.3.m2.4.5.3.3.2" xref="S3.p2.3.m2.4.5.3.3.2.cmml"><mi id="S3.p2.3.m2.4.5.3.3.2.2" xref="S3.p2.3.m2.4.5.3.3.2.2.cmml">C</mi><mrow id="S3.p2.3.m2.4.4.2.4" xref="S3.p2.3.m2.4.4.2.3.cmml"><mi id="S3.p2.3.m2.3.3.1.1" xref="S3.p2.3.m2.3.3.1.1.cmml">i</mi><mo id="S3.p2.3.m2.4.4.2.4.1" xref="S3.p2.3.m2.4.4.2.3.cmml">,</mo><mi id="S3.p2.3.m2.4.4.2.2" xref="S3.p2.3.m2.4.4.2.2.cmml">j</mi></mrow></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.3.m2.4b"><apply id="S3.p2.3.m2.4.5.cmml" xref="S3.p2.3.m2.4.5"><eq id="S3.p2.3.m2.4.5.1.cmml" xref="S3.p2.3.m2.4.5.1"></eq><ci id="S3.p2.3.m2.4.5.2.cmml" xref="S3.p2.3.m2.4.5.2">𝐸</ci><apply id="S3.p2.3.m2.4.5.3.cmml" xref="S3.p2.3.m2.4.5.3"><times id="S3.p2.3.m2.4.5.3.1.cmml" xref="S3.p2.3.m2.4.5.3.1"></times><apply id="S3.p2.3.m2.4.5.3.2.cmml" xref="S3.p2.3.m2.4.5.3.2"><divide id="S3.p2.3.m2.4.5.3.2.1.cmml" xref="S3.p2.3.m2.4.5.3.2"></divide><cn id="S3.p2.3.m2.4.5.3.2.2.cmml" type="integer" xref="S3.p2.3.m2.4.5.3.2.2">1</cn><cn id="S3.p2.3.m2.4.5.3.2.3.cmml" type="integer" xref="S3.p2.3.m2.4.5.3.2.3">2</cn></apply><apply id="S3.p2.3.m2.4.5.3.3.cmml" xref="S3.p2.3.m2.4.5.3.3"><apply id="S3.p2.3.m2.4.5.3.3.1.cmml" xref="S3.p2.3.m2.4.5.3.3.1"><csymbol cd="ambiguous" id="S3.p2.3.m2.4.5.3.3.1.1.cmml" xref="S3.p2.3.m2.4.5.3.3.1">superscript</csymbol><apply id="S3.p2.3.m2.4.5.3.3.1.2.cmml" xref="S3.p2.3.m2.4.5.3.3.1"><csymbol cd="ambiguous" id="S3.p2.3.m2.4.5.3.3.1.2.1.cmml" xref="S3.p2.3.m2.4.5.3.3.1">subscript</csymbol><sum id="S3.p2.3.m2.4.5.3.3.1.2.2.cmml" xref="S3.p2.3.m2.4.5.3.3.1.2.2"></sum><apply id="S3.p2.3.m2.2.2.2.cmml" xref="S3.p2.3.m2.2.2.2"><eq id="S3.p2.3.m2.2.2.2.3.cmml" xref="S3.p2.3.m2.2.2.2.3"></eq><list id="S3.p2.3.m2.2.2.2.4.1.cmml" xref="S3.p2.3.m2.2.2.2.4.2"><ci id="S3.p2.3.m2.1.1.1.1.cmml" xref="S3.p2.3.m2.1.1.1.1">𝑖</ci><ci id="S3.p2.3.m2.2.2.2.2.cmml" xref="S3.p2.3.m2.2.2.2.2">𝑗</ci></list><cn id="S3.p2.3.m2.2.2.2.5.cmml" type="integer" xref="S3.p2.3.m2.2.2.2.5">1</cn></apply></apply><ci id="S3.p2.3.m2.4.5.3.3.1.3.cmml" xref="S3.p2.3.m2.4.5.3.3.1.3">𝑉</ci></apply><apply id="S3.p2.3.m2.4.5.3.3.2.cmml" xref="S3.p2.3.m2.4.5.3.3.2"><csymbol cd="ambiguous" id="S3.p2.3.m2.4.5.3.3.2.1.cmml" xref="S3.p2.3.m2.4.5.3.3.2">subscript</csymbol><ci id="S3.p2.3.m2.4.5.3.3.2.2.cmml" xref="S3.p2.3.m2.4.5.3.3.2.2">𝐶</ci><list id="S3.p2.3.m2.4.4.2.3.cmml" xref="S3.p2.3.m2.4.4.2.4"><ci id="S3.p2.3.m2.3.3.1.1.cmml" xref="S3.p2.3.m2.3.3.1.1">𝑖</ci><ci id="S3.p2.3.m2.4.4.2.2.cmml" xref="S3.p2.3.m2.4.4.2.2">𝑗</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m2.4c">E=\frac{1}{2}\sum_{i,j=1}^{V}C_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m2.4d">italic_E = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.12">The coordinate <math alttext="x_{i,j}" class="ltx_Math" display="inline" id="S3.p3.1.m1.2"><semantics id="S3.p3.1.m1.2a"><msub id="S3.p3.1.m1.2.3" xref="S3.p3.1.m1.2.3.cmml"><mi id="S3.p3.1.m1.2.3.2" xref="S3.p3.1.m1.2.3.2.cmml">x</mi><mrow id="S3.p3.1.m1.2.2.2.4" xref="S3.p3.1.m1.2.2.2.3.cmml"><mi id="S3.p3.1.m1.1.1.1.1" xref="S3.p3.1.m1.1.1.1.1.cmml">i</mi><mo id="S3.p3.1.m1.2.2.2.4.1" xref="S3.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.p3.1.m1.2.2.2.2" xref="S3.p3.1.m1.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.2b"><apply id="S3.p3.1.m1.2.3.cmml" xref="S3.p3.1.m1.2.3"><csymbol cd="ambiguous" id="S3.p3.1.m1.2.3.1.cmml" xref="S3.p3.1.m1.2.3">subscript</csymbol><ci id="S3.p3.1.m1.2.3.2.cmml" xref="S3.p3.1.m1.2.3.2">𝑥</ci><list id="S3.p3.1.m1.2.2.2.3.cmml" xref="S3.p3.1.m1.2.2.2.4"><ci id="S3.p3.1.m1.1.1.1.1.cmml" xref="S3.p3.1.m1.1.1.1.1">𝑖</ci><ci id="S3.p3.1.m1.2.2.2.2.cmml" xref="S3.p3.1.m1.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.2c">x_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.2d">italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> on the edge <math alttext="(i,j)" class="ltx_Math" display="inline" id="S3.p3.2.m2.2"><semantics id="S3.p3.2.m2.2a"><mrow id="S3.p3.2.m2.2.3.2" xref="S3.p3.2.m2.2.3.1.cmml"><mo id="S3.p3.2.m2.2.3.2.1" stretchy="false" xref="S3.p3.2.m2.2.3.1.cmml">(</mo><mi id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml">i</mi><mo id="S3.p3.2.m2.2.3.2.2" xref="S3.p3.2.m2.2.3.1.cmml">,</mo><mi id="S3.p3.2.m2.2.2" xref="S3.p3.2.m2.2.2.cmml">j</mi><mo id="S3.p3.2.m2.2.3.2.3" stretchy="false" xref="S3.p3.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.2b"><interval closure="open" id="S3.p3.2.m2.2.3.1.cmml" xref="S3.p3.2.m2.2.3.2"><ci id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1">𝑖</ci><ci id="S3.p3.2.m2.2.2.cmml" xref="S3.p3.2.m2.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.2d">( italic_i , italic_j )</annotation></semantics></math> is assigned as <math alttext="[0,L_{i,j}]" class="ltx_Math" display="inline" id="S3.p3.3.m3.4"><semantics id="S3.p3.3.m3.4a"><mrow id="S3.p3.3.m3.4.4.1" xref="S3.p3.3.m3.4.4.2.cmml"><mo id="S3.p3.3.m3.4.4.1.2" stretchy="false" xref="S3.p3.3.m3.4.4.2.cmml">[</mo><mn id="S3.p3.3.m3.3.3" xref="S3.p3.3.m3.3.3.cmml">0</mn><mo id="S3.p3.3.m3.4.4.1.3" xref="S3.p3.3.m3.4.4.2.cmml">,</mo><msub id="S3.p3.3.m3.4.4.1.1" xref="S3.p3.3.m3.4.4.1.1.cmml"><mi id="S3.p3.3.m3.4.4.1.1.2" xref="S3.p3.3.m3.4.4.1.1.2.cmml">L</mi><mrow id="S3.p3.3.m3.2.2.2.4" xref="S3.p3.3.m3.2.2.2.3.cmml"><mi id="S3.p3.3.m3.1.1.1.1" xref="S3.p3.3.m3.1.1.1.1.cmml">i</mi><mo id="S3.p3.3.m3.2.2.2.4.1" xref="S3.p3.3.m3.2.2.2.3.cmml">,</mo><mi id="S3.p3.3.m3.2.2.2.2" xref="S3.p3.3.m3.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.p3.3.m3.4.4.1.4" stretchy="false" xref="S3.p3.3.m3.4.4.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.4b"><interval closure="closed" id="S3.p3.3.m3.4.4.2.cmml" xref="S3.p3.3.m3.4.4.1"><cn id="S3.p3.3.m3.3.3.cmml" type="integer" xref="S3.p3.3.m3.3.3">0</cn><apply id="S3.p3.3.m3.4.4.1.1.cmml" xref="S3.p3.3.m3.4.4.1.1"><csymbol cd="ambiguous" id="S3.p3.3.m3.4.4.1.1.1.cmml" xref="S3.p3.3.m3.4.4.1.1">subscript</csymbol><ci id="S3.p3.3.m3.4.4.1.1.2.cmml" xref="S3.p3.3.m3.4.4.1.1.2">𝐿</ci><list id="S3.p3.3.m3.2.2.2.3.cmml" xref="S3.p3.3.m3.2.2.2.4"><ci id="S3.p3.3.m3.1.1.1.1.cmml" xref="S3.p3.3.m3.1.1.1.1">𝑖</ci><ci id="S3.p3.3.m3.2.2.2.2.cmml" xref="S3.p3.3.m3.2.2.2.2">𝑗</ci></list></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.4c">[0,L_{i,j}]</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.4d">[ 0 , italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ]</annotation></semantics></math> for all <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S3.p3.4.m4.2"><semantics id="S3.p3.4.m4.2a"><mrow id="S3.p3.4.m4.2.3" xref="S3.p3.4.m4.2.3.cmml"><msub id="S3.p3.4.m4.2.3.2" xref="S3.p3.4.m4.2.3.2.cmml"><mi id="S3.p3.4.m4.2.3.2.2" xref="S3.p3.4.m4.2.3.2.2.cmml">C</mi><mrow id="S3.p3.4.m4.2.2.2.4" xref="S3.p3.4.m4.2.2.2.3.cmml"><mi id="S3.p3.4.m4.1.1.1.1" xref="S3.p3.4.m4.1.1.1.1.cmml">i</mi><mo id="S3.p3.4.m4.2.2.2.4.1" xref="S3.p3.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.p3.4.m4.2.2.2.2" xref="S3.p3.4.m4.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.p3.4.m4.2.3.1" xref="S3.p3.4.m4.2.3.1.cmml">=</mo><mn id="S3.p3.4.m4.2.3.3" xref="S3.p3.4.m4.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.4.m4.2b"><apply id="S3.p3.4.m4.2.3.cmml" xref="S3.p3.4.m4.2.3"><eq id="S3.p3.4.m4.2.3.1.cmml" xref="S3.p3.4.m4.2.3.1"></eq><apply id="S3.p3.4.m4.2.3.2.cmml" xref="S3.p3.4.m4.2.3.2"><csymbol cd="ambiguous" id="S3.p3.4.m4.2.3.2.1.cmml" xref="S3.p3.4.m4.2.3.2">subscript</csymbol><ci id="S3.p3.4.m4.2.3.2.2.cmml" xref="S3.p3.4.m4.2.3.2.2">𝐶</ci><list id="S3.p3.4.m4.2.2.2.3.cmml" xref="S3.p3.4.m4.2.2.2.4"><ci id="S3.p3.4.m4.1.1.1.1.cmml" xref="S3.p3.4.m4.1.1.1.1">𝑖</ci><ci id="S3.p3.4.m4.2.2.2.2.cmml" xref="S3.p3.4.m4.2.2.2.2">𝑗</ci></list></apply><cn id="S3.p3.4.m4.2.3.3.cmml" type="integer" xref="S3.p3.4.m4.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="i<j" class="ltx_Math" display="inline" id="S3.p3.5.m5.1"><semantics id="S3.p3.5.m5.1a"><mrow id="S3.p3.5.m5.1.1" xref="S3.p3.5.m5.1.1.cmml"><mi id="S3.p3.5.m5.1.1.2" xref="S3.p3.5.m5.1.1.2.cmml">i</mi><mo id="S3.p3.5.m5.1.1.1" xref="S3.p3.5.m5.1.1.1.cmml"><</mo><mi id="S3.p3.5.m5.1.1.3" xref="S3.p3.5.m5.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.5.m5.1b"><apply id="S3.p3.5.m5.1.1.cmml" xref="S3.p3.5.m5.1.1"><lt id="S3.p3.5.m5.1.1.1.cmml" xref="S3.p3.5.m5.1.1.1"></lt><ci id="S3.p3.5.m5.1.1.2.cmml" xref="S3.p3.5.m5.1.1.2">𝑖</ci><ci id="S3.p3.5.m5.1.1.3.cmml" xref="S3.p3.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m5.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m5.1d">italic_i < italic_j</annotation></semantics></math>, where <math alttext="L_{i,j}" class="ltx_Math" display="inline" id="S3.p3.6.m6.2"><semantics id="S3.p3.6.m6.2a"><msub id="S3.p3.6.m6.2.3" xref="S3.p3.6.m6.2.3.cmml"><mi id="S3.p3.6.m6.2.3.2" xref="S3.p3.6.m6.2.3.2.cmml">L</mi><mrow id="S3.p3.6.m6.2.2.2.4" xref="S3.p3.6.m6.2.2.2.3.cmml"><mi id="S3.p3.6.m6.1.1.1.1" xref="S3.p3.6.m6.1.1.1.1.cmml">i</mi><mo id="S3.p3.6.m6.2.2.2.4.1" xref="S3.p3.6.m6.2.2.2.3.cmml">,</mo><mi id="S3.p3.6.m6.2.2.2.2" xref="S3.p3.6.m6.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p3.6.m6.2b"><apply id="S3.p3.6.m6.2.3.cmml" xref="S3.p3.6.m6.2.3"><csymbol cd="ambiguous" id="S3.p3.6.m6.2.3.1.cmml" xref="S3.p3.6.m6.2.3">subscript</csymbol><ci id="S3.p3.6.m6.2.3.2.cmml" xref="S3.p3.6.m6.2.3.2">𝐿</ci><list id="S3.p3.6.m6.2.2.2.3.cmml" xref="S3.p3.6.m6.2.2.2.4"><ci id="S3.p3.6.m6.1.1.1.1.cmml" xref="S3.p3.6.m6.1.1.1.1">𝑖</ci><ci id="S3.p3.6.m6.2.2.2.2.cmml" xref="S3.p3.6.m6.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.6.m6.2c">L_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.6.m6.2d">italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is the length of the edge <math alttext="(i,j)" class="ltx_Math" display="inline" id="S3.p3.7.m7.2"><semantics id="S3.p3.7.m7.2a"><mrow id="S3.p3.7.m7.2.3.2" xref="S3.p3.7.m7.2.3.1.cmml"><mo id="S3.p3.7.m7.2.3.2.1" stretchy="false" xref="S3.p3.7.m7.2.3.1.cmml">(</mo><mi id="S3.p3.7.m7.1.1" xref="S3.p3.7.m7.1.1.cmml">i</mi><mo id="S3.p3.7.m7.2.3.2.2" xref="S3.p3.7.m7.2.3.1.cmml">,</mo><mi id="S3.p3.7.m7.2.2" xref="S3.p3.7.m7.2.2.cmml">j</mi><mo id="S3.p3.7.m7.2.3.2.3" stretchy="false" xref="S3.p3.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.7.m7.2b"><interval closure="open" id="S3.p3.7.m7.2.3.1.cmml" xref="S3.p3.7.m7.2.3.2"><ci id="S3.p3.7.m7.1.1.cmml" xref="S3.p3.7.m7.1.1">𝑖</ci><ci id="S3.p3.7.m7.2.2.cmml" xref="S3.p3.7.m7.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.7.m7.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.7.m7.2d">( italic_i , italic_j )</annotation></semantics></math>. The wavefunction <math alttext="\Psi" class="ltx_Math" display="inline" id="S3.p3.8.m8.1"><semantics id="S3.p3.8.m8.1a"><mi id="S3.p3.8.m8.1.1" mathvariant="normal" xref="S3.p3.8.m8.1.1.cmml">Ψ</mi><annotation-xml encoding="MathML-Content" id="S3.p3.8.m8.1b"><ci id="S3.p3.8.m8.1.1.cmml" xref="S3.p3.8.m8.1.1">Ψ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.8.m8.1c">\Psi</annotation><annotation encoding="application/x-llamapun" id="S3.p3.8.m8.1d">roman_Ψ</annotation></semantics></math> is a vector function with <math alttext="E" class="ltx_Math" display="inline" id="S3.p3.9.m9.1"><semantics id="S3.p3.9.m9.1a"><mi id="S3.p3.9.m9.1.1" xref="S3.p3.9.m9.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S3.p3.9.m9.1b"><ci id="S3.p3.9.m9.1.1.cmml" xref="S3.p3.9.m9.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.9.m9.1c">E</annotation><annotation encoding="application/x-llamapun" id="S3.p3.9.m9.1d">italic_E</annotation></semantics></math> components, where each component is defined as <math alttext="\Psi_{i,j}(x_{i,j})" class="ltx_Math" display="inline" id="S3.p3.10.m10.5"><semantics id="S3.p3.10.m10.5a"><mrow id="S3.p3.10.m10.5.5" xref="S3.p3.10.m10.5.5.cmml"><msub id="S3.p3.10.m10.5.5.3" xref="S3.p3.10.m10.5.5.3.cmml"><mi id="S3.p3.10.m10.5.5.3.2" mathvariant="normal" xref="S3.p3.10.m10.5.5.3.2.cmml">Ψ</mi><mrow id="S3.p3.10.m10.2.2.2.4" xref="S3.p3.10.m10.2.2.2.3.cmml"><mi id="S3.p3.10.m10.1.1.1.1" xref="S3.p3.10.m10.1.1.1.1.cmml">i</mi><mo id="S3.p3.10.m10.2.2.2.4.1" xref="S3.p3.10.m10.2.2.2.3.cmml">,</mo><mi id="S3.p3.10.m10.2.2.2.2" xref="S3.p3.10.m10.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.p3.10.m10.5.5.2" xref="S3.p3.10.m10.5.5.2.cmml">⁢</mo><mrow id="S3.p3.10.m10.5.5.1.1" xref="S3.p3.10.m10.5.5.1.1.1.cmml"><mo id="S3.p3.10.m10.5.5.1.1.2" stretchy="false" xref="S3.p3.10.m10.5.5.1.1.1.cmml">(</mo><msub id="S3.p3.10.m10.5.5.1.1.1" xref="S3.p3.10.m10.5.5.1.1.1.cmml"><mi id="S3.p3.10.m10.5.5.1.1.1.2" xref="S3.p3.10.m10.5.5.1.1.1.2.cmml">x</mi><mrow id="S3.p3.10.m10.4.4.2.4" xref="S3.p3.10.m10.4.4.2.3.cmml"><mi id="S3.p3.10.m10.3.3.1.1" xref="S3.p3.10.m10.3.3.1.1.cmml">i</mi><mo id="S3.p3.10.m10.4.4.2.4.1" xref="S3.p3.10.m10.4.4.2.3.cmml">,</mo><mi id="S3.p3.10.m10.4.4.2.2" xref="S3.p3.10.m10.4.4.2.2.cmml">j</mi></mrow></msub><mo id="S3.p3.10.m10.5.5.1.1.3" stretchy="false" xref="S3.p3.10.m10.5.5.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.10.m10.5b"><apply id="S3.p3.10.m10.5.5.cmml" xref="S3.p3.10.m10.5.5"><times id="S3.p3.10.m10.5.5.2.cmml" xref="S3.p3.10.m10.5.5.2"></times><apply id="S3.p3.10.m10.5.5.3.cmml" xref="S3.p3.10.m10.5.5.3"><csymbol cd="ambiguous" id="S3.p3.10.m10.5.5.3.1.cmml" xref="S3.p3.10.m10.5.5.3">subscript</csymbol><ci id="S3.p3.10.m10.5.5.3.2.cmml" xref="S3.p3.10.m10.5.5.3.2">Ψ</ci><list id="S3.p3.10.m10.2.2.2.3.cmml" xref="S3.p3.10.m10.2.2.2.4"><ci id="S3.p3.10.m10.1.1.1.1.cmml" xref="S3.p3.10.m10.1.1.1.1">𝑖</ci><ci id="S3.p3.10.m10.2.2.2.2.cmml" xref="S3.p3.10.m10.2.2.2.2">𝑗</ci></list></apply><apply id="S3.p3.10.m10.5.5.1.1.1.cmml" xref="S3.p3.10.m10.5.5.1.1"><csymbol cd="ambiguous" id="S3.p3.10.m10.5.5.1.1.1.1.cmml" xref="S3.p3.10.m10.5.5.1.1">subscript</csymbol><ci id="S3.p3.10.m10.5.5.1.1.1.2.cmml" xref="S3.p3.10.m10.5.5.1.1.1.2">𝑥</ci><list id="S3.p3.10.m10.4.4.2.3.cmml" xref="S3.p3.10.m10.4.4.2.4"><ci id="S3.p3.10.m10.3.3.1.1.cmml" xref="S3.p3.10.m10.3.3.1.1">𝑖</ci><ci id="S3.p3.10.m10.4.4.2.2.cmml" xref="S3.p3.10.m10.4.4.2.2">𝑗</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.10.m10.5c">\Psi_{i,j}(x_{i,j})</annotation><annotation encoding="application/x-llamapun" id="S3.p3.10.m10.5d">roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S3.p3.11.m11.2"><semantics id="S3.p3.11.m11.2a"><mrow id="S3.p3.11.m11.2.3" xref="S3.p3.11.m11.2.3.cmml"><msub id="S3.p3.11.m11.2.3.2" xref="S3.p3.11.m11.2.3.2.cmml"><mi id="S3.p3.11.m11.2.3.2.2" xref="S3.p3.11.m11.2.3.2.2.cmml">C</mi><mrow id="S3.p3.11.m11.2.2.2.4" xref="S3.p3.11.m11.2.2.2.3.cmml"><mi id="S3.p3.11.m11.1.1.1.1" xref="S3.p3.11.m11.1.1.1.1.cmml">i</mi><mo id="S3.p3.11.m11.2.2.2.4.1" xref="S3.p3.11.m11.2.2.2.3.cmml">,</mo><mi id="S3.p3.11.m11.2.2.2.2" xref="S3.p3.11.m11.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.p3.11.m11.2.3.1" xref="S3.p3.11.m11.2.3.1.cmml">=</mo><mn id="S3.p3.11.m11.2.3.3" xref="S3.p3.11.m11.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.11.m11.2b"><apply id="S3.p3.11.m11.2.3.cmml" xref="S3.p3.11.m11.2.3"><eq id="S3.p3.11.m11.2.3.1.cmml" xref="S3.p3.11.m11.2.3.1"></eq><apply id="S3.p3.11.m11.2.3.2.cmml" xref="S3.p3.11.m11.2.3.2"><csymbol cd="ambiguous" id="S3.p3.11.m11.2.3.2.1.cmml" xref="S3.p3.11.m11.2.3.2">subscript</csymbol><ci id="S3.p3.11.m11.2.3.2.2.cmml" xref="S3.p3.11.m11.2.3.2.2">𝐶</ci><list id="S3.p3.11.m11.2.2.2.3.cmml" xref="S3.p3.11.m11.2.2.2.4"><ci id="S3.p3.11.m11.1.1.1.1.cmml" xref="S3.p3.11.m11.1.1.1.1">𝑖</ci><ci id="S3.p3.11.m11.2.2.2.2.cmml" xref="S3.p3.11.m11.2.2.2.2">𝑗</ci></list></apply><cn id="S3.p3.11.m11.2.3.3.cmml" type="integer" xref="S3.p3.11.m11.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.11.m11.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.p3.11.m11.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>. The Schrödinger equation on each edge of the graph is written as (in units <math alttext="\hbar=2m=1" class="ltx_Math" display="inline" id="S3.p3.12.m12.1"><semantics id="S3.p3.12.m12.1a"><mrow id="S3.p3.12.m12.1.1" xref="S3.p3.12.m12.1.1.cmml"><mi id="S3.p3.12.m12.1.1.2" mathvariant="normal" xref="S3.p3.12.m12.1.1.2.cmml">ℏ</mi><mo id="S3.p3.12.m12.1.1.3" xref="S3.p3.12.m12.1.1.3.cmml">=</mo><mrow id="S3.p3.12.m12.1.1.4" xref="S3.p3.12.m12.1.1.4.cmml"><mn id="S3.p3.12.m12.1.1.4.2" xref="S3.p3.12.m12.1.1.4.2.cmml">2</mn><mo id="S3.p3.12.m12.1.1.4.1" xref="S3.p3.12.m12.1.1.4.1.cmml">⁢</mo><mi id="S3.p3.12.m12.1.1.4.3" xref="S3.p3.12.m12.1.1.4.3.cmml">m</mi></mrow><mo id="S3.p3.12.m12.1.1.5" xref="S3.p3.12.m12.1.1.5.cmml">=</mo><mn id="S3.p3.12.m12.1.1.6" xref="S3.p3.12.m12.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.12.m12.1b"><apply id="S3.p3.12.m12.1.1.cmml" xref="S3.p3.12.m12.1.1"><and id="S3.p3.12.m12.1.1a.cmml" xref="S3.p3.12.m12.1.1"></and><apply id="S3.p3.12.m12.1.1b.cmml" xref="S3.p3.12.m12.1.1"><eq id="S3.p3.12.m12.1.1.3.cmml" xref="S3.p3.12.m12.1.1.3"></eq><csymbol cd="latexml" id="S3.p3.12.m12.1.1.2.cmml" xref="S3.p3.12.m12.1.1.2">Planck-constant-over-2-pi</csymbol><apply id="S3.p3.12.m12.1.1.4.cmml" xref="S3.p3.12.m12.1.1.4"><times id="S3.p3.12.m12.1.1.4.1.cmml" xref="S3.p3.12.m12.1.1.4.1"></times><cn id="S3.p3.12.m12.1.1.4.2.cmml" type="integer" xref="S3.p3.12.m12.1.1.4.2">2</cn><ci id="S3.p3.12.m12.1.1.4.3.cmml" xref="S3.p3.12.m12.1.1.4.3">𝑚</ci></apply></apply><apply id="S3.p3.12.m12.1.1c.cmml" xref="S3.p3.12.m12.1.1"><eq id="S3.p3.12.m12.1.1.5.cmml" xref="S3.p3.12.m12.1.1.5"></eq><share href="https://arxiv.org/html/2411.14397v1#S3.p3.12.m12.1.1.4.cmml" id="S3.p3.12.m12.1.1d.cmml" xref="S3.p3.12.m12.1.1"></share><cn id="S3.p3.12.m12.1.1.6.cmml" type="integer" xref="S3.p3.12.m12.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.12.m12.1c">\hbar=2m=1</annotation><annotation encoding="application/x-llamapun" id="S3.p3.12.m12.1d">roman_ℏ = 2 italic_m = 1</annotation></semantics></math>)</p> <table class="ltx_equation ltx_eqn_table" id="S3.E20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{d^{2}\Psi_{i,j}(x_{i,j})}{dx^{2}}+k^{2}\Psi_{i,j}(x_{i,j})=0." class="ltx_Math" display="block" id="S3.E20.m1.10"><semantics id="S3.E20.m1.10a"><mrow id="S3.E20.m1.10.10.1" xref="S3.E20.m1.10.10.1.1.cmml"><mrow id="S3.E20.m1.10.10.1.1" xref="S3.E20.m1.10.10.1.1.cmml"><mrow id="S3.E20.m1.10.10.1.1.1" xref="S3.E20.m1.10.10.1.1.1.cmml"><mfrac id="S3.E20.m1.5.5" xref="S3.E20.m1.5.5.cmml"><mrow id="S3.E20.m1.5.5.5" xref="S3.E20.m1.5.5.5.cmml"><msup id="S3.E20.m1.5.5.5.7" xref="S3.E20.m1.5.5.5.7.cmml"><mi id="S3.E20.m1.5.5.5.7.2" xref="S3.E20.m1.5.5.5.7.2.cmml">d</mi><mn id="S3.E20.m1.5.5.5.7.3" xref="S3.E20.m1.5.5.5.7.3.cmml">2</mn></msup><mo id="S3.E20.m1.5.5.5.6" xref="S3.E20.m1.5.5.5.6.cmml">⁢</mo><msub id="S3.E20.m1.5.5.5.8" xref="S3.E20.m1.5.5.5.8.cmml"><mi id="S3.E20.m1.5.5.5.8.2" mathvariant="normal" xref="S3.E20.m1.5.5.5.8.2.cmml">Ψ</mi><mrow id="S3.E20.m1.2.2.2.2.2.4" xref="S3.E20.m1.2.2.2.2.2.3.cmml"><mi id="S3.E20.m1.1.1.1.1.1.1" xref="S3.E20.m1.1.1.1.1.1.1.cmml">i</mi><mo id="S3.E20.m1.2.2.2.2.2.4.1" xref="S3.E20.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S3.E20.m1.2.2.2.2.2.2" xref="S3.E20.m1.2.2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.E20.m1.5.5.5.6a" xref="S3.E20.m1.5.5.5.6.cmml">⁢</mo><mrow id="S3.E20.m1.5.5.5.5.1" xref="S3.E20.m1.5.5.5.5.1.1.cmml"><mo id="S3.E20.m1.5.5.5.5.1.2" stretchy="false" xref="S3.E20.m1.5.5.5.5.1.1.cmml">(</mo><msub id="S3.E20.m1.5.5.5.5.1.1" xref="S3.E20.m1.5.5.5.5.1.1.cmml"><mi id="S3.E20.m1.5.5.5.5.1.1.2" xref="S3.E20.m1.5.5.5.5.1.1.2.cmml">x</mi><mrow id="S3.E20.m1.4.4.4.4.2.4" xref="S3.E20.m1.4.4.4.4.2.3.cmml"><mi id="S3.E20.m1.3.3.3.3.1.1" 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wise on each edge of the graph</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Psi_{i,j}(x_{i,j})\mapsto-\frac{d^{2}\Psi_{i,j}(x_{i,j})}{dx^{2}}." class="ltx_Math" display="block" id="S3.Ex11.m1.10"><semantics id="S3.Ex11.m1.10a"><mrow id="S3.Ex11.m1.10.10.1" xref="S3.Ex11.m1.10.10.1.1.cmml"><mrow id="S3.Ex11.m1.10.10.1.1" xref="S3.Ex11.m1.10.10.1.1.cmml"><mrow id="S3.Ex11.m1.10.10.1.1.1" xref="S3.Ex11.m1.10.10.1.1.1.cmml"><msub id="S3.Ex11.m1.10.10.1.1.1.3" xref="S3.Ex11.m1.10.10.1.1.1.3.cmml"><mi id="S3.Ex11.m1.10.10.1.1.1.3.2" mathvariant="normal" xref="S3.Ex11.m1.10.10.1.1.1.3.2.cmml">Ψ</mi><mrow id="S3.Ex11.m1.2.2.2.4" xref="S3.Ex11.m1.2.2.2.3.cmml"><mi id="S3.Ex11.m1.1.1.1.1" xref="S3.Ex11.m1.1.1.1.1.cmml">i</mi><mo id="S3.Ex11.m1.2.2.2.4.1" xref="S3.Ex11.m1.2.2.2.3.cmml">,</mo><mi 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encoding="application/x-tex" id="S3.Ex11.m1.10c">\Psi_{i,j}(x_{i,j})\mapsto-\frac{d^{2}\Psi_{i,j}(x_{i,j})}{dx^{2}}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex11.m1.10d">roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ↦ - divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG italic_d italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.15">Usually, one associates boundary conditions to the operator <math alttext="H" class="ltx_Math" display="inline" id="S3.p3.14.m1.1"><semantics id="S3.p3.14.m1.1a"><mi id="S3.p3.14.m1.1.1" xref="S3.p3.14.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.p3.14.m1.1b"><ci id="S3.p3.14.m1.1.1.cmml" xref="S3.p3.14.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.14.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.p3.14.m1.1d">italic_H</annotation></semantics></math>. Without any boundary conditions, the Schrödinger operator <math alttext="H" class="ltx_Math" display="inline" id="S3.p3.15.m2.1"><semantics id="S3.p3.15.m2.1a"><mi id="S3.p3.15.m2.1.1" xref="S3.p3.15.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.p3.15.m2.1b"><ci id="S3.p3.15.m2.1.1.cmml" xref="S3.p3.15.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.15.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.p3.15.m2.1d">italic_H</annotation></semantics></math> satisfies</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx16"> <tbody id="S3.E21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}\langle H\Psi,\Phi\rangle-\langle\Psi,&H\Phi\rangle=% \\ \sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{V}&C_{i,j}\left[\Psi_{i,j}(x_{i,j})\frac{d\Phi_{i,j}^{*}(x% _{i,j})}{dx}\right.\\ &\left.-\Phi_{i,j}^{*}(x_{i,j})\frac{d\Psi_{i,j}(x_{i,j})}{dx}\right]\bigg{|}_% {x_{i,j}=0}^{x_{i,j}=L_{i,j}}.\end{split}" class="ltx_Math" display="inline" id="S3.E21.m1.42"><semantics id="S3.E21.m1.42a"><mtable columnspacing="0pt" id="S3.E21.m1.41.41a" rowspacing="0pt"><mtr id="S3.E21.m1.41.41aa"><mtd class="ltx_align_right" columnalign="right" id="S3.E21.m1.41.41ab"><mrow id="S3.E21.m1.11.11.11.10.10a"><mrow id="S3.E21.m1.11.11.11.10.10a.11"><mo id="S3.E21.m1.2.2.2.1.1.1" stretchy="false" xref="S3.E21.m1.42.42.1.1.1.cmml">⟨</mo><mi id="S3.E21.m1.3.3.3.2.2.2" xref="S3.E21.m1.3.3.3.2.2.2.cmml">H</mi><mi id="S3.E21.m1.4.4.4.3.3.3" mathvariant="normal" xref="S3.E21.m1.4.4.4.3.3.3.cmml">Ψ</mi><mo id="S3.E21.m1.5.5.5.4.4.4" xref="S3.E21.m1.42.42.1.1.1.cmml">,</mo><mi id="S3.E21.m1.6.6.6.5.5.5" mathvariant="normal" xref="S3.E21.m1.6.6.6.5.5.5.cmml">Φ</mi><mo id="S3.E21.m1.7.7.7.6.6.6" stretchy="false" xref="S3.E21.m1.42.42.1.1.1.cmml">⟩</mo></mrow><mo id="S3.E21.m1.8.8.8.7.7.7" xref="S3.E21.m1.8.8.8.7.7.7.cmml">−</mo><mrow id="S3.E21.m1.11.11.11.10.10a.12"><mo id="S3.E21.m1.9.9.9.8.8.8" stretchy="false" xref="S3.E21.m1.42.42.1.1.1.cmml">⟨</mo><mi id="S3.E21.m1.10.10.10.9.9.9" mathvariant="normal" xref="S3.E21.m1.10.10.10.9.9.9.cmml">Ψ</mi><mo id="S3.E21.m1.11.11.11.10.10.10" xref="S3.E21.m1.42.42.1.1.1.cmml">,</mo></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.E21.m1.41.41ac"><mrow id="S3.E21.m1.15.15.15.14.4a"><mi id="S3.E21.m1.12.12.12.11.1.1" xref="S3.E21.m1.12.12.12.11.1.1.cmml">H</mi><mi id="S3.E21.m1.13.13.13.12.2.2" mathvariant="normal" xref="S3.E21.m1.13.13.13.12.2.2.cmml">Φ</mi><mo id="S3.E21.m1.14.14.14.13.3.3" stretchy="false" xref="S3.E21.m1.42.42.1.1.1.cmml">⟩</mo><mo id="S3.E21.m1.15.15.15.14.4.4" xref="S3.E21.m1.15.15.15.14.4.4.cmml">=</mo></mrow></mtd></mtr><mtr id="S3.E21.m1.41.41ad"><mtd class="ltx_align_right" 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xref="S3.E21.m1.40.40.40.13.13.13.1.4.2.4"><ci id="S3.E21.m1.40.40.40.13.13.13.1.3.1.1.cmml" xref="S3.E21.m1.40.40.40.13.13.13.1.3.1.1">𝑖</ci><ci id="S3.E21.m1.40.40.40.13.13.13.1.4.2.2.cmml" xref="S3.E21.m1.40.40.40.13.13.13.1.4.2.2">𝑗</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E21.m1.42c">\displaystyle\begin{split}\langle H\Psi,\Phi\rangle-\langle\Psi,&H\Phi\rangle=% \\ \sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{V}&C_{i,j}\left[\Psi_{i,j}(x_{i,j})\frac{d\Phi_{i,j}^{*}(x% _{i,j})}{dx}\right.\\ &\left.-\Phi_{i,j}^{*}(x_{i,j})\frac{d\Psi_{i,j}(x_{i,j})}{dx}\right]\bigg{|}_% {x_{i,j}=0}^{x_{i,j}=L_{i,j}}.\end{split}</annotation><annotation encoding="application/x-llamapun" id="S3.E21.m1.42d">start_ROW start_CELL ⟨ italic_H roman_Ψ , roman_Φ ⟩ - ⟨ roman_Ψ , end_CELL start_CELL italic_H roman_Φ ⟩ = end_CELL end_ROW start_ROW start_CELL ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_i , italic_j = 1 end_CELL end_ROW start_ROW start_CELL italic_i < italic_j end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [ roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) divide start_ARG italic_d roman_Φ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG italic_d italic_x end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - roman_Φ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) divide start_ARG italic_d roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG italic_d italic_x end_ARG ] | start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.18">Obviously, whenever the Schrödinger operator <math alttext="H" class="ltx_Math" display="inline" id="S3.p3.16.m1.1"><semantics id="S3.p3.16.m1.1a"><mi id="S3.p3.16.m1.1.1" xref="S3.p3.16.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.p3.16.m1.1b"><ci id="S3.p3.16.m1.1.1.cmml" xref="S3.p3.16.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.16.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.p3.16.m1.1d">italic_H</annotation></semantics></math> is self-adjoint, the expression in (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E21" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">21</span></a>) is zero. E.g., this is the case when the functions from the domain satisfy continuity on the graph. That is, for each vertex there exists a complex number <math alttext="\phi_{i}" class="ltx_Math" display="inline" id="S3.p3.17.m2.1"><semantics id="S3.p3.17.m2.1a"><msub id="S3.p3.17.m2.1.1" xref="S3.p3.17.m2.1.1.cmml"><mi id="S3.p3.17.m2.1.1.2" xref="S3.p3.17.m2.1.1.2.cmml">ϕ</mi><mi id="S3.p3.17.m2.1.1.3" xref="S3.p3.17.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.17.m2.1b"><apply id="S3.p3.17.m2.1.1.cmml" xref="S3.p3.17.m2.1.1"><csymbol cd="ambiguous" id="S3.p3.17.m2.1.1.1.cmml" xref="S3.p3.17.m2.1.1">subscript</csymbol><ci id="S3.p3.17.m2.1.1.2.cmml" xref="S3.p3.17.m2.1.1.2">italic-ϕ</ci><ci id="S3.p3.17.m2.1.1.3.cmml" xref="S3.p3.17.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.17.m2.1c">\phi_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.17.m2.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="i\in\{1,\ldots,V\}" 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id="S3.E22.m1.13.13.3.3.1.1.cmml" xref="S3.E22.m1.13.13.3.3.1.1">𝑖</ci><ci id="S3.E22.m1.14.14.4.4.2.2.cmml" xref="S3.E22.m1.14.14.4.4.2.2">𝑗</ci></list></apply></apply></apply><apply id="S3.E22.m1.15.15.1.1.2.2.3.cmml" xref="S3.E22.m1.15.15.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E22.m1.15.15.1.1.2.2.3.1.cmml" xref="S3.E22.m1.15.15.1.1.2.2.3">subscript</csymbol><ci id="S3.E22.m1.15.15.1.1.2.2.3.2.cmml" xref="S3.E22.m1.15.15.1.1.2.2.3.2">italic-ϕ</ci><ci id="S3.E22.m1.15.15.1.1.2.2.3.3.cmml" xref="S3.E22.m1.15.15.1.1.2.2.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E22.m1.15c">\Psi_{i,j}(x_{i,j})\big{|}_{x_{i,j}=0}=\phi_{i},\quad\Psi_{i,j}(x_{i,j})\big{|% }_{x_{i,j}=L_{i,j}}=\phi_{j},</annotation><annotation encoding="application/x-llamapun" id="S3.E22.m1.15d">roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.21">for <math alttext="i,j" class="ltx_Math" display="inline" id="S3.p3.19.m1.2"><semantics id="S3.p3.19.m1.2a"><mrow id="S3.p3.19.m1.2.3.2" xref="S3.p3.19.m1.2.3.1.cmml"><mi id="S3.p3.19.m1.1.1" xref="S3.p3.19.m1.1.1.cmml">i</mi><mo id="S3.p3.19.m1.2.3.2.1" xref="S3.p3.19.m1.2.3.1.cmml">,</mo><mi id="S3.p3.19.m1.2.2" xref="S3.p3.19.m1.2.2.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.19.m1.2b"><list id="S3.p3.19.m1.2.3.1.cmml" xref="S3.p3.19.m1.2.3.2"><ci id="S3.p3.19.m1.1.1.cmml" xref="S3.p3.19.m1.1.1">𝑖</ci><ci id="S3.p3.19.m1.2.2.cmml" xref="S3.p3.19.m1.2.2">𝑗</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.19.m1.2c">i,j</annotation><annotation encoding="application/x-llamapun" id="S3.p3.19.m1.2d">italic_i , italic_j</annotation></semantics></math> such that <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S3.p3.20.m2.2"><semantics id="S3.p3.20.m2.2a"><mrow id="S3.p3.20.m2.2.3" xref="S3.p3.20.m2.2.3.cmml"><msub id="S3.p3.20.m2.2.3.2" xref="S3.p3.20.m2.2.3.2.cmml"><mi id="S3.p3.20.m2.2.3.2.2" xref="S3.p3.20.m2.2.3.2.2.cmml">C</mi><mrow id="S3.p3.20.m2.2.2.2.4" xref="S3.p3.20.m2.2.2.2.3.cmml"><mi id="S3.p3.20.m2.1.1.1.1" xref="S3.p3.20.m2.1.1.1.1.cmml">i</mi><mo id="S3.p3.20.m2.2.2.2.4.1" xref="S3.p3.20.m2.2.2.2.3.cmml">,</mo><mi id="S3.p3.20.m2.2.2.2.2" xref="S3.p3.20.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.p3.20.m2.2.3.1" xref="S3.p3.20.m2.2.3.1.cmml">=</mo><mn id="S3.p3.20.m2.2.3.3" xref="S3.p3.20.m2.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.20.m2.2b"><apply id="S3.p3.20.m2.2.3.cmml" xref="S3.p3.20.m2.2.3"><eq id="S3.p3.20.m2.2.3.1.cmml" xref="S3.p3.20.m2.2.3.1"></eq><apply id="S3.p3.20.m2.2.3.2.cmml" xref="S3.p3.20.m2.2.3.2"><csymbol cd="ambiguous" id="S3.p3.20.m2.2.3.2.1.cmml" xref="S3.p3.20.m2.2.3.2">subscript</csymbol><ci id="S3.p3.20.m2.2.3.2.2.cmml" xref="S3.p3.20.m2.2.3.2.2">𝐶</ci><list id="S3.p3.20.m2.2.2.2.3.cmml" xref="S3.p3.20.m2.2.2.2.4"><ci id="S3.p3.20.m2.1.1.1.1.cmml" xref="S3.p3.20.m2.1.1.1.1">𝑖</ci><ci id="S3.p3.20.m2.2.2.2.2.cmml" xref="S3.p3.20.m2.2.2.2.2">𝑗</ci></list></apply><cn id="S3.p3.20.m2.2.3.3.cmml" type="integer" xref="S3.p3.20.m2.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.20.m2.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.p3.20.m2.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>, <math alttext="i<j" class="ltx_Math" display="inline" id="S3.p3.21.m3.1"><semantics id="S3.p3.21.m3.1a"><mrow id="S3.p3.21.m3.1.1" xref="S3.p3.21.m3.1.1.cmml"><mi id="S3.p3.21.m3.1.1.2" xref="S3.p3.21.m3.1.1.2.cmml">i</mi><mo id="S3.p3.21.m3.1.1.1" xref="S3.p3.21.m3.1.1.1.cmml"><</mo><mi id="S3.p3.21.m3.1.1.3" xref="S3.p3.21.m3.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.21.m3.1b"><apply id="S3.p3.21.m3.1.1.cmml" xref="S3.p3.21.m3.1.1"><lt id="S3.p3.21.m3.1.1.1.cmml" xref="S3.p3.21.m3.1.1.1"></lt><ci id="S3.p3.21.m3.1.1.2.cmml" xref="S3.p3.21.m3.1.1.2">𝑖</ci><ci id="S3.p3.21.m3.1.1.3.cmml" xref="S3.p3.21.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.21.m3.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S3.p3.21.m3.1d">italic_i < italic_j</annotation></semantics></math>, and some current conservation rule <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib6" title="">6</a>]</cite>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx17"> <tbody id="S3.E23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}-\sum_{j<i}C_{i,j}&\frac{d\Psi_{j,i}(x_{j,i})}{dx}% \bigg{|}_{x_{j,i}=L_{j,i}}\\ &+\sum_{j>i}C_{i,j}\frac{d\Psi_{i,j}(x_{i,j})}{dx}\bigg{|}_{x_{i,j}=0}=\lambda% _{i}\phi_{i},\end{split}" 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id="S3.E23.m1.23.23.1.1.1.4.2.cmml" xref="S3.E23.m1.5.5.5.5.5aa"><csymbol cd="ambiguous" id="S3.E23.m1.23.23.1.1.1.4.2.1.cmml" xref="S3.E23.m1.5.5.5.5.5aa">subscript</csymbol><ci id="S3.E23.m1.18.18.18.10.10.10.cmml" xref="S3.E23.m1.18.18.18.10.10.10">𝜆</ci><ci id="S3.E23.m1.19.19.19.11.11.11.1.cmml" xref="S3.E23.m1.19.19.19.11.11.11.1">𝑖</ci></apply><apply id="S3.E23.m1.23.23.1.1.1.4.3.cmml" xref="S3.E23.m1.5.5.5.5.5aa"><csymbol cd="ambiguous" id="S3.E23.m1.23.23.1.1.1.4.3.1.cmml" xref="S3.E23.m1.5.5.5.5.5aa">subscript</csymbol><ci id="S3.E23.m1.20.20.20.12.12.12.cmml" xref="S3.E23.m1.20.20.20.12.12.12">italic-ϕ</ci><ci id="S3.E23.m1.21.21.21.13.13.13.1.cmml" xref="S3.E23.m1.21.21.21.13.13.13.1">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E23.m1.24c">\displaystyle\begin{split}-\sum_{j<i}C_{i,j}&\frac{d\Psi_{j,i}(x_{j,i})}{dx}% \bigg{|}_{x_{j,i}=L_{j,i}}\\ &+\sum_{j>i}C_{i,j}\frac{d\Psi_{i,j}(x_{i,j})}{dx}\bigg{|}_{x_{i,j}=0}=\lambda% _{i}\phi_{i},\end{split}</annotation><annotation encoding="application/x-llamapun" id="S3.E23.m1.24d">start_ROW start_CELL - ∑ start_POSTSUBSCRIPT italic_j < italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_CELL start_CELL divide start_ARG italic_d roman_Ψ start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG italic_d italic_x end_ARG | start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_j > italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT divide start_ARG italic_d roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG italic_d italic_x end_ARG | start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.26">for some real parameters <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S3.p3.22.m1.1"><semantics id="S3.p3.22.m1.1a"><msub id="S3.p3.22.m1.1.1" xref="S3.p3.22.m1.1.1.cmml"><mi id="S3.p3.22.m1.1.1.2" xref="S3.p3.22.m1.1.1.2.cmml">λ</mi><mi id="S3.p3.22.m1.1.1.3" xref="S3.p3.22.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.22.m1.1b"><apply id="S3.p3.22.m1.1.1.cmml" xref="S3.p3.22.m1.1.1"><csymbol cd="ambiguous" id="S3.p3.22.m1.1.1.1.cmml" xref="S3.p3.22.m1.1.1">subscript</csymbol><ci id="S3.p3.22.m1.1.1.2.cmml" xref="S3.p3.22.m1.1.1.2">𝜆</ci><ci id="S3.p3.22.m1.1.1.3.cmml" xref="S3.p3.22.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.22.m1.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.22.m1.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i" class="ltx_Math" display="inline" id="S3.p3.23.m2.1"><semantics id="S3.p3.23.m2.1a"><mi id="S3.p3.23.m2.1.1" xref="S3.p3.23.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p3.23.m2.1b"><ci id="S3.p3.23.m2.1.1.cmml" xref="S3.p3.23.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.23.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p3.23.m2.1d">italic_i</annotation></semantics></math> in <math alttext="\{1,...,V\}" class="ltx_Math" display="inline" id="S3.p3.24.m3.3"><semantics id="S3.p3.24.m3.3a"><mrow id="S3.p3.24.m3.3.4.2" xref="S3.p3.24.m3.3.4.1.cmml"><mo id="S3.p3.24.m3.3.4.2.1" stretchy="false" xref="S3.p3.24.m3.3.4.1.cmml">{</mo><mn id="S3.p3.24.m3.1.1" xref="S3.p3.24.m3.1.1.cmml">1</mn><mo id="S3.p3.24.m3.3.4.2.2" xref="S3.p3.24.m3.3.4.1.cmml">,</mo><mi id="S3.p3.24.m3.2.2" mathvariant="normal" xref="S3.p3.24.m3.2.2.cmml">…</mi><mo id="S3.p3.24.m3.3.4.2.3" xref="S3.p3.24.m3.3.4.1.cmml">,</mo><mi id="S3.p3.24.m3.3.3" xref="S3.p3.24.m3.3.3.cmml">V</mi><mo id="S3.p3.24.m3.3.4.2.4" stretchy="false" xref="S3.p3.24.m3.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.24.m3.3b"><set id="S3.p3.24.m3.3.4.1.cmml" xref="S3.p3.24.m3.3.4.2"><cn id="S3.p3.24.m3.1.1.cmml" type="integer" xref="S3.p3.24.m3.1.1">1</cn><ci id="S3.p3.24.m3.2.2.cmml" xref="S3.p3.24.m3.2.2">…</ci><ci id="S3.p3.24.m3.3.3.cmml" xref="S3.p3.24.m3.3.3">𝑉</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.24.m3.3c">\{1,...,V\}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.24.m3.3d">{ 1 , … , italic_V }</annotation></semantics></math>. Note that the case <math alttext="\lambda_{i}=0" class="ltx_Math" display="inline" id="S3.p3.25.m4.1"><semantics id="S3.p3.25.m4.1a"><mrow id="S3.p3.25.m4.1.1" xref="S3.p3.25.m4.1.1.cmml"><msub id="S3.p3.25.m4.1.1.2" xref="S3.p3.25.m4.1.1.2.cmml"><mi id="S3.p3.25.m4.1.1.2.2" xref="S3.p3.25.m4.1.1.2.2.cmml">λ</mi><mi id="S3.p3.25.m4.1.1.2.3" xref="S3.p3.25.m4.1.1.2.3.cmml">i</mi></msub><mo id="S3.p3.25.m4.1.1.1" xref="S3.p3.25.m4.1.1.1.cmml">=</mo><mn id="S3.p3.25.m4.1.1.3" xref="S3.p3.25.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.25.m4.1b"><apply id="S3.p3.25.m4.1.1.cmml" xref="S3.p3.25.m4.1.1"><eq id="S3.p3.25.m4.1.1.1.cmml" xref="S3.p3.25.m4.1.1.1"></eq><apply id="S3.p3.25.m4.1.1.2.cmml" xref="S3.p3.25.m4.1.1.2"><csymbol cd="ambiguous" id="S3.p3.25.m4.1.1.2.1.cmml" xref="S3.p3.25.m4.1.1.2">subscript</csymbol><ci id="S3.p3.25.m4.1.1.2.2.cmml" xref="S3.p3.25.m4.1.1.2.2">𝜆</ci><ci id="S3.p3.25.m4.1.1.2.3.cmml" xref="S3.p3.25.m4.1.1.2.3">𝑖</ci></apply><cn id="S3.p3.25.m4.1.1.3.cmml" type="integer" xref="S3.p3.25.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.25.m4.1c">\lambda_{i}=0</annotation><annotation encoding="application/x-llamapun" id="S3.p3.25.m4.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0</annotation></semantics></math> for all <math alttext="i\text{ in }\{1,...,V\}" class="ltx_Math" display="inline" id="S3.p3.26.m5.3"><semantics id="S3.p3.26.m5.3a"><mrow id="S3.p3.26.m5.3.4" xref="S3.p3.26.m5.3.4.cmml"><mi id="S3.p3.26.m5.3.4.2" xref="S3.p3.26.m5.3.4.2.cmml">i</mi><mo id="S3.p3.26.m5.3.4.1" xref="S3.p3.26.m5.3.4.1.cmml">⁢</mo><mtext id="S3.p3.26.m5.3.4.3" xref="S3.p3.26.m5.3.4.3a.cmml"> in </mtext><mo id="S3.p3.26.m5.3.4.1a" xref="S3.p3.26.m5.3.4.1.cmml">⁢</mo><mrow id="S3.p3.26.m5.3.4.4.2" xref="S3.p3.26.m5.3.4.4.1.cmml"><mo id="S3.p3.26.m5.3.4.4.2.1" stretchy="false" xref="S3.p3.26.m5.3.4.4.1.cmml">{</mo><mn id="S3.p3.26.m5.1.1" xref="S3.p3.26.m5.1.1.cmml">1</mn><mo id="S3.p3.26.m5.3.4.4.2.2" xref="S3.p3.26.m5.3.4.4.1.cmml">,</mo><mi id="S3.p3.26.m5.2.2" mathvariant="normal" xref="S3.p3.26.m5.2.2.cmml">…</mi><mo id="S3.p3.26.m5.3.4.4.2.3" xref="S3.p3.26.m5.3.4.4.1.cmml">,</mo><mi id="S3.p3.26.m5.3.3" xref="S3.p3.26.m5.3.3.cmml">V</mi><mo id="S3.p3.26.m5.3.4.4.2.4" stretchy="false" xref="S3.p3.26.m5.3.4.4.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.26.m5.3b"><apply id="S3.p3.26.m5.3.4.cmml" xref="S3.p3.26.m5.3.4"><times id="S3.p3.26.m5.3.4.1.cmml" xref="S3.p3.26.m5.3.4.1"></times><ci id="S3.p3.26.m5.3.4.2.cmml" xref="S3.p3.26.m5.3.4.2">𝑖</ci><ci id="S3.p3.26.m5.3.4.3a.cmml" xref="S3.p3.26.m5.3.4.3"><mtext id="S3.p3.26.m5.3.4.3.cmml" xref="S3.p3.26.m5.3.4.3"> in </mtext></ci><set id="S3.p3.26.m5.3.4.4.1.cmml" xref="S3.p3.26.m5.3.4.4.2"><cn id="S3.p3.26.m5.1.1.cmml" type="integer" xref="S3.p3.26.m5.1.1">1</cn><ci id="S3.p3.26.m5.2.2.cmml" xref="S3.p3.26.m5.2.2">…</ci><ci id="S3.p3.26.m5.3.3.cmml" xref="S3.p3.26.m5.3.3">𝑉</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.26.m5.3c">i\text{ in }\{1,...,V\}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.26.m5.3d">italic_i in { 1 , … , italic_V }</annotation></semantics></math> is called Kirchhoff rule.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.7">The solution of the Schrödinger equation on the graph in (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E20" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">20</span></a>) can be written as</p> <table class="ltx_equationgroup ltx_eqn_gather ltx_eqn_table" id="S6.EGx18"> <tbody id="S3.E24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\displaystyle\Psi_{i,j}(x_{i,j})=C_{i,j}\left[A_{i,j}\sin[k(L_{i,j}-x_{i,j})]+% B_{i,j}\sin(kx_{i,j})\right]," class="ltx_Math" display="block" id="S3.E24.m1.19"><semantics id="S3.E24.m1.19a"><mrow id="S3.E24.m1.19.19.1" xref="S3.E24.m1.19.19.1.1.cmml"><mrow id="S3.E24.m1.19.19.1.1" xref="S3.E24.m1.19.19.1.1.cmml"><mrow id="S3.E24.m1.19.19.1.1.1" xref="S3.E24.m1.19.19.1.1.1.cmml"><msub id="S3.E24.m1.19.19.1.1.1.3" xref="S3.E24.m1.19.19.1.1.1.3.cmml"><mi id="S3.E24.m1.19.19.1.1.1.3.2" mathvariant="normal" xref="S3.E24.m1.19.19.1.1.1.3.2.cmml">Ψ</mi><mrow id="S3.E24.m1.2.2.2.4" xref="S3.E24.m1.2.2.2.3.cmml"><mi id="S3.E24.m1.1.1.1.1" xref="S3.E24.m1.1.1.1.1.cmml">i</mi><mo id="S3.E24.m1.2.2.2.4.1" xref="S3.E24.m1.2.2.2.3.cmml">,</mo><mi id="S3.E24.m1.2.2.2.2" xref="S3.E24.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.E24.m1.19.19.1.1.1.2" xref="S3.E24.m1.19.19.1.1.1.2.cmml">⁢</mo><mrow id="S3.E24.m1.19.19.1.1.1.1.1" 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xref="S3.E24.m1.16.16.2.2">𝑗</ci></list></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E24.m1.19c">\displaystyle\Psi_{i,j}(x_{i,j})=C_{i,j}\left[A_{i,j}\sin[k(L_{i,j}-x_{i,j})]+% B_{i,j}\sin(kx_{i,j})\right],</annotation><annotation encoding="application/x-llamapun" id="S3.E24.m1.19d">roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) = italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [ italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_sin [ italic_k ( italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ] + italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_sin ( italic_k italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.1">for all <math alttext="i<j" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mrow id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml"><mi id="S3.p4.1.m1.1.1.2" xref="S3.p4.1.m1.1.1.2.cmml">i</mi><mo id="S3.p4.1.m1.1.1.1" xref="S3.p4.1.m1.1.1.1.cmml"><</mo><mi id="S3.p4.1.m1.1.1.3" xref="S3.p4.1.m1.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><apply id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1"><lt id="S3.p4.1.m1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1"></lt><ci id="S3.p4.1.m1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.2">𝑖</ci><ci id="S3.p4.1.m1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_i < italic_j</annotation></semantics></math>. Substituting the solution (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E24" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">24</span></a>) into the vertex boundary conditions (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E22" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">22</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E23" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">23</span></a>) leads to the secular equations as</p> <table class="ltx_equation ltx_eqn_table" id="S3.E25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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id="S3.E25.m1.33.33.1.1.1.2.2.3.3.1.cmml">subscript</csymbol><ci id="S3.E25.m1.30.30.30.14.14.14.cmml" xref="S3.E25.m1.30.30.30.14.14.14">italic-ϕ</ci><ci id="S3.E25.m1.31.31.31.15.15.15.1.cmml" xref="S3.E25.m1.31.31.31.15.15.15.1">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E25.m1.35c">\begin{split}&C_{i,j}A_{i,j}\sin(kL_{i,j})=C_{i,j}\phi_{i},\\ &C_{i,j}B_{i,j}\sin(kL_{i,j})=C_{i,j}\phi_{j},\end{split}</annotation><annotation encoding="application/x-llamapun" id="S3.E25.m1.35d">start_ROW start_CELL end_CELL start_CELL italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_sin ( italic_k italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) = italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_sin ( italic_k italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) = italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.4">for <math alttext="i,j" class="ltx_Math" display="inline" id="S3.p4.2.m1.2"><semantics id="S3.p4.2.m1.2a"><mrow id="S3.p4.2.m1.2.3.2" xref="S3.p4.2.m1.2.3.1.cmml"><mi id="S3.p4.2.m1.1.1" xref="S3.p4.2.m1.1.1.cmml">i</mi><mo id="S3.p4.2.m1.2.3.2.1" xref="S3.p4.2.m1.2.3.1.cmml">,</mo><mi id="S3.p4.2.m1.2.2" xref="S3.p4.2.m1.2.2.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.2.m1.2b"><list id="S3.p4.2.m1.2.3.1.cmml" xref="S3.p4.2.m1.2.3.2"><ci id="S3.p4.2.m1.1.1.cmml" xref="S3.p4.2.m1.1.1">𝑖</ci><ci id="S3.p4.2.m1.2.2.cmml" xref="S3.p4.2.m1.2.2">𝑗</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m1.2c">i,j</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m1.2d">italic_i , italic_j</annotation></semantics></math> such that <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S3.p4.3.m2.2"><semantics id="S3.p4.3.m2.2a"><mrow id="S3.p4.3.m2.2.3" xref="S3.p4.3.m2.2.3.cmml"><msub id="S3.p4.3.m2.2.3.2" xref="S3.p4.3.m2.2.3.2.cmml"><mi id="S3.p4.3.m2.2.3.2.2" xref="S3.p4.3.m2.2.3.2.2.cmml">C</mi><mrow id="S3.p4.3.m2.2.2.2.4" xref="S3.p4.3.m2.2.2.2.3.cmml"><mi id="S3.p4.3.m2.1.1.1.1" xref="S3.p4.3.m2.1.1.1.1.cmml">i</mi><mo id="S3.p4.3.m2.2.2.2.4.1" xref="S3.p4.3.m2.2.2.2.3.cmml">,</mo><mi id="S3.p4.3.m2.2.2.2.2" xref="S3.p4.3.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.p4.3.m2.2.3.1" xref="S3.p4.3.m2.2.3.1.cmml">=</mo><mn id="S3.p4.3.m2.2.3.3" xref="S3.p4.3.m2.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.3.m2.2b"><apply id="S3.p4.3.m2.2.3.cmml" xref="S3.p4.3.m2.2.3"><eq id="S3.p4.3.m2.2.3.1.cmml" xref="S3.p4.3.m2.2.3.1"></eq><apply id="S3.p4.3.m2.2.3.2.cmml" xref="S3.p4.3.m2.2.3.2"><csymbol cd="ambiguous" id="S3.p4.3.m2.2.3.2.1.cmml" xref="S3.p4.3.m2.2.3.2">subscript</csymbol><ci id="S3.p4.3.m2.2.3.2.2.cmml" xref="S3.p4.3.m2.2.3.2.2">𝐶</ci><list id="S3.p4.3.m2.2.2.2.3.cmml" xref="S3.p4.3.m2.2.2.2.4"><ci id="S3.p4.3.m2.1.1.1.1.cmml" xref="S3.p4.3.m2.1.1.1.1">𝑖</ci><ci id="S3.p4.3.m2.2.2.2.2.cmml" xref="S3.p4.3.m2.2.2.2.2">𝑗</ci></list></apply><cn id="S3.p4.3.m2.2.3.3.cmml" type="integer" xref="S3.p4.3.m2.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m2.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m2.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="i<j" class="ltx_Math" display="inline" id="S3.p4.4.m3.1"><semantics id="S3.p4.4.m3.1a"><mrow id="S3.p4.4.m3.1.1" xref="S3.p4.4.m3.1.1.cmml"><mi id="S3.p4.4.m3.1.1.2" xref="S3.p4.4.m3.1.1.2.cmml">i</mi><mo id="S3.p4.4.m3.1.1.1" xref="S3.p4.4.m3.1.1.1.cmml"><</mo><mi id="S3.p4.4.m3.1.1.3" xref="S3.p4.4.m3.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.4.m3.1b"><apply id="S3.p4.4.m3.1.1.cmml" xref="S3.p4.4.m3.1.1"><lt id="S3.p4.4.m3.1.1.1.cmml" xref="S3.p4.4.m3.1.1.1"></lt><ci id="S3.p4.4.m3.1.1.2.cmml" xref="S3.p4.4.m3.1.1.2">𝑖</ci><ci id="S3.p4.4.m3.1.1.3.cmml" xref="S3.p4.4.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m3.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m3.1d">italic_i < italic_j</annotation></semantics></math>, and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx19"> <tbody id="S3.E26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}\sum_{j<i}C_{i,j}&k\left[A_{j,i}-B_{j,i}\cos(kL_{j,i% })\right]\\ +&\sum_{j>i}C_{i,j}k\left[-A_{i,j}\cos(kL_{i,j})+B_{i,j}\right]=\lambda_{i}% \phi_{i},\end{split}" class="ltx_Math" display="inline" id="S3.E26.m1.47"><semantics id="S3.E26.m1.47a"><mtable columnspacing="0pt" id="S3.E26.m1.47.47.3" rowspacing="0pt"><mtr id="S3.E26.m1.47.47.3a"><mtd class="ltx_align_right" columnalign="right" id="S3.E26.m1.47.47.3b"><mrow id="S3.E26.m1.4.4.4.4.4a"><mstyle displaystyle="true" id="S3.E26.m1.4.4.4.4.4a.5"><munder id="S3.E26.m1.4.4.4.4.4a.5a"><mo id="S3.E26.m1.1.1.1.1.1.1" movablelimits="false" xref="S3.E26.m1.1.1.1.1.1.1.cmml">∑</mo><mrow 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+&\sum_{j>i}C_{i,j}k\left[-A_{i,j}\cos(kL_{i,j})+B_{i,j}\right]=\lambda_{i}% \phi_{i},\end{split}</annotation><annotation encoding="application/x-llamapun" id="S3.E26.m1.47d">start_ROW start_CELL ∑ start_POSTSUBSCRIPT italic_j < italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_CELL start_CELL italic_k [ italic_A start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT - italic_B start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT roman_cos ( italic_k italic_L start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT ) ] end_CELL end_ROW start_ROW start_CELL + end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_j > italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_k [ - italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_cos ( italic_k italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) + italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ] = italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.6">for all <math alttext="i" class="ltx_Math" display="inline" id="S3.p4.5.m1.1"><semantics id="S3.p4.5.m1.1a"><mi id="S3.p4.5.m1.1.1" xref="S3.p4.5.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p4.5.m1.1b"><ci id="S3.p4.5.m1.1.1.cmml" xref="S3.p4.5.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.5.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p4.5.m1.1d">italic_i</annotation></semantics></math> in <math alttext="\{1,...,V\}" class="ltx_Math" display="inline" id="S3.p4.6.m2.3"><semantics id="S3.p4.6.m2.3a"><mrow id="S3.p4.6.m2.3.4.2" xref="S3.p4.6.m2.3.4.1.cmml"><mo id="S3.p4.6.m2.3.4.2.1" stretchy="false" xref="S3.p4.6.m2.3.4.1.cmml">{</mo><mn id="S3.p4.6.m2.1.1" xref="S3.p4.6.m2.1.1.cmml">1</mn><mo id="S3.p4.6.m2.3.4.2.2" xref="S3.p4.6.m2.3.4.1.cmml">,</mo><mi id="S3.p4.6.m2.2.2" mathvariant="normal" xref="S3.p4.6.m2.2.2.cmml">…</mi><mo id="S3.p4.6.m2.3.4.2.3" xref="S3.p4.6.m2.3.4.1.cmml">,</mo><mi id="S3.p4.6.m2.3.3" xref="S3.p4.6.m2.3.3.cmml">V</mi><mo id="S3.p4.6.m2.3.4.2.4" stretchy="false" xref="S3.p4.6.m2.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.6.m2.3b"><set id="S3.p4.6.m2.3.4.1.cmml" xref="S3.p4.6.m2.3.4.2"><cn id="S3.p4.6.m2.1.1.cmml" type="integer" xref="S3.p4.6.m2.1.1">1</cn><ci id="S3.p4.6.m2.2.2.cmml" xref="S3.p4.6.m2.2.2">…</ci><ci id="S3.p4.6.m2.3.3.cmml" xref="S3.p4.6.m2.3.3">𝑉</ci></set></annotation-xml><annotation encoding="application/x-tex" 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xref="S3.p5.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><apply id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p5.2.m2.1.1.1.cmml" xref="S3.p5.2.m2.1.1">subscript</csymbol><ci id="S3.p5.2.m2.1.1.2.cmml" xref="S3.p5.2.m2.1.1.2">italic-ϕ</ci><ci id="S3.p5.2.m2.1.1.3.cmml" xref="S3.p5.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">\phi_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and has a non-trivial solution only when</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx20"> <tbody id="S3.E27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\det(M(k))=0," class="ltx_Math" display="inline" id="S3.E27.m1.2"><semantics id="S3.E27.m1.2a"><mrow id="S3.E27.m1.2.2.1" xref="S3.E27.m1.2.2.1.1.cmml"><mrow id="S3.E27.m1.2.2.1.1" xref="S3.E27.m1.2.2.1.1.cmml"><mrow id="S3.E27.m1.2.2.1.1.1" xref="S3.E27.m1.2.2.1.1.1.cmml"><mo id="S3.E27.m1.2.2.1.1.1.2" movablelimits="false" rspace="0em" xref="S3.E27.m1.2.2.1.1.1.2.cmml">det</mo><mrow id="S3.E27.m1.2.2.1.1.1.1.1" xref="S3.E27.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S3.E27.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.E27.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E27.m1.2.2.1.1.1.1.1.1" xref="S3.E27.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S3.E27.m1.2.2.1.1.1.1.1.1.2" xref="S3.E27.m1.2.2.1.1.1.1.1.1.2.cmml">M</mi><mo id="S3.E27.m1.2.2.1.1.1.1.1.1.1" xref="S3.E27.m1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.E27.m1.2.2.1.1.1.1.1.1.3.2" xref="S3.E27.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S3.E27.m1.2.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.E27.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mi id="S3.E27.m1.1.1" xref="S3.E27.m1.1.1.cmml">k</mi><mo id="S3.E27.m1.2.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.E27.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E27.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S3.E27.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E27.m1.2.2.1.1.2" xref="S3.E27.m1.2.2.1.1.2.cmml">=</mo><mn id="S3.E27.m1.2.2.1.1.3" xref="S3.E27.m1.2.2.1.1.3.cmml">0</mn></mrow><mo id="S3.E27.m1.2.2.1.2" xref="S3.E27.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E27.m1.2b"><apply id="S3.E27.m1.2.2.1.1.cmml" xref="S3.E27.m1.2.2.1"><eq id="S3.E27.m1.2.2.1.1.2.cmml" xref="S3.E27.m1.2.2.1.1.2"></eq><apply id="S3.E27.m1.2.2.1.1.1.cmml" xref="S3.E27.m1.2.2.1.1.1"><determinant id="S3.E27.m1.2.2.1.1.1.2.cmml" xref="S3.E27.m1.2.2.1.1.1.2"></determinant><apply id="S3.E27.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E27.m1.2.2.1.1.1.1.1"><times id="S3.E27.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.E27.m1.2.2.1.1.1.1.1.1.1"></times><ci id="S3.E27.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.E27.m1.2.2.1.1.1.1.1.1.2">𝑀</ci><ci id="S3.E27.m1.1.1.cmml" xref="S3.E27.m1.1.1">𝑘</ci></apply></apply><cn id="S3.E27.m1.2.2.1.1.3.cmml" type="integer" xref="S3.E27.m1.2.2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E27.m1.2c">\displaystyle\det(M(k))=0,</annotation><annotation encoding="application/x-llamapun" id="S3.E27.m1.2d">roman_det ( italic_M ( italic_k ) ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.4">where <math alttext="M(k)" class="ltx_Math" display="inline" id="S3.p5.3.m1.1"><semantics id="S3.p5.3.m1.1a"><mrow id="S3.p5.3.m1.1.2" xref="S3.p5.3.m1.1.2.cmml"><mi id="S3.p5.3.m1.1.2.2" xref="S3.p5.3.m1.1.2.2.cmml">M</mi><mo id="S3.p5.3.m1.1.2.1" xref="S3.p5.3.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p5.3.m1.1.2.3.2" xref="S3.p5.3.m1.1.2.cmml"><mo id="S3.p5.3.m1.1.2.3.2.1" stretchy="false" xref="S3.p5.3.m1.1.2.cmml">(</mo><mi id="S3.p5.3.m1.1.1" xref="S3.p5.3.m1.1.1.cmml">k</mi><mo id="S3.p5.3.m1.1.2.3.2.2" stretchy="false" xref="S3.p5.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.3.m1.1b"><apply id="S3.p5.3.m1.1.2.cmml" xref="S3.p5.3.m1.1.2"><times id="S3.p5.3.m1.1.2.1.cmml" xref="S3.p5.3.m1.1.2.1"></times><ci id="S3.p5.3.m1.1.2.2.cmml" xref="S3.p5.3.m1.1.2.2">𝑀</ci><ci id="S3.p5.3.m1.1.1.cmml" xref="S3.p5.3.m1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m1.1c">M(k)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m1.1d">italic_M ( italic_k )</annotation></semantics></math> is a matrix where all the coefficients of the linear equations (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E25" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">25</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E26" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">26</span></a>) are collected. It depends on the eigenvalue parameter <math alttext="k" class="ltx_Math" display="inline" id="S3.p5.4.m2.1"><semantics id="S3.p5.4.m2.1a"><mi id="S3.p5.4.m2.1.1" xref="S3.p5.4.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.p5.4.m2.1b"><ci id="S3.p5.4.m2.1.1.cmml" xref="S3.p5.4.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m2.1d">italic_k</annotation></semantics></math> and its size is</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(2E+V)\times(2E+V)." class="ltx_Math" display="block" id="S3.Ex12.m1.1"><semantics id="S3.Ex12.m1.1a"><mrow id="S3.Ex12.m1.1.1.1" xref="S3.Ex12.m1.1.1.1.1.cmml"><mrow id="S3.Ex12.m1.1.1.1.1" xref="S3.Ex12.m1.1.1.1.1.cmml"><mrow 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id="S3.Ex12.m1.1.1.1.1.cmml" xref="S3.Ex12.m1.1.1.1"><times id="S3.Ex12.m1.1.1.1.1.3.cmml" xref="S3.Ex12.m1.1.1.1.1.3"></times><apply id="S3.Ex12.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex12.m1.1.1.1.1.1.1"><plus id="S3.Ex12.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex12.m1.1.1.1.1.1.1.1.1"></plus><apply id="S3.Ex12.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex12.m1.1.1.1.1.1.1.1.2"><times id="S3.Ex12.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex12.m1.1.1.1.1.1.1.1.2.1"></times><cn id="S3.Ex12.m1.1.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.Ex12.m1.1.1.1.1.1.1.1.2.2">2</cn><ci id="S3.Ex12.m1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Ex12.m1.1.1.1.1.1.1.1.2.3">𝐸</ci></apply><ci id="S3.Ex12.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex12.m1.1.1.1.1.1.1.1.3">𝑉</ci></apply><apply id="S3.Ex12.m1.1.1.1.1.2.1.1.cmml" xref="S3.Ex12.m1.1.1.1.1.2.1"><plus id="S3.Ex12.m1.1.1.1.1.2.1.1.1.cmml" xref="S3.Ex12.m1.1.1.1.1.2.1.1.1"></plus><apply id="S3.Ex12.m1.1.1.1.1.2.1.1.2.cmml" xref="S3.Ex12.m1.1.1.1.1.2.1.1.2"><times id="S3.Ex12.m1.1.1.1.1.2.1.1.2.1.cmml" xref="S3.Ex12.m1.1.1.1.1.2.1.1.2.1"></times><cn id="S3.Ex12.m1.1.1.1.1.2.1.1.2.2.cmml" type="integer" xref="S3.Ex12.m1.1.1.1.1.2.1.1.2.2">2</cn><ci id="S3.Ex12.m1.1.1.1.1.2.1.1.2.3.cmml" xref="S3.Ex12.m1.1.1.1.1.2.1.1.2.3">𝐸</ci></apply><ci id="S3.Ex12.m1.1.1.1.1.2.1.1.3.cmml" xref="S3.Ex12.m1.1.1.1.1.2.1.1.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex12.m1.1c">(2E+V)\times(2E+V).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex12.m1.1d">( 2 italic_E + italic_V ) × ( 2 italic_E + italic_V ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Branched lattices</h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Arbitrary branching topology</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Adhering to the formulation of the quantum graph problem proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib6" title="">6</a>]</cite>, in this subsection, we present a prescription for solving the problem for discrete quantum graphs.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.6">The discrete graph is a set of <math alttext="V" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_V</annotation></semantics></math> vertices connected by <math alttext="E" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_E</annotation></semantics></math> discrete edges. We denote the edge which connects the vertices <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mi id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">italic_i</annotation></semantics></math> and <math alttext="j" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mi id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_j</annotation></semantics></math> as <math alttext="(i,j)" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.2"><semantics id="S4.SS1.p2.5.m5.2a"><mrow id="S4.SS1.p2.5.m5.2.3.2" xref="S4.SS1.p2.5.m5.2.3.1.cmml"><mo id="S4.SS1.p2.5.m5.2.3.2.1" stretchy="false" xref="S4.SS1.p2.5.m5.2.3.1.cmml">(</mo><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">i</mi><mo id="S4.SS1.p2.5.m5.2.3.2.2" xref="S4.SS1.p2.5.m5.2.3.1.cmml">,</mo><mi id="S4.SS1.p2.5.m5.2.2" xref="S4.SS1.p2.5.m5.2.2.cmml">j</mi><mo id="S4.SS1.p2.5.m5.2.3.2.3" stretchy="false" xref="S4.SS1.p2.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.2b"><interval closure="open" id="S4.SS1.p2.5.m5.2.3.1.cmml" xref="S4.SS1.p2.5.m5.2.3.2"><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑖</ci><ci id="S4.SS1.p2.5.m5.2.2.cmml" xref="S4.SS1.p2.5.m5.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.2d">( italic_i , italic_j )</annotation></semantics></math> for <math alttext="i<j" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mrow id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml"><mi id="S4.SS1.p2.6.m6.1.1.2" xref="S4.SS1.p2.6.m6.1.1.2.cmml">i</mi><mo id="S4.SS1.p2.6.m6.1.1.1" xref="S4.SS1.p2.6.m6.1.1.1.cmml"><</mo><mi id="S4.SS1.p2.6.m6.1.1.3" xref="S4.SS1.p2.6.m6.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><apply id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1"><lt id="S4.SS1.p2.6.m6.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1"></lt><ci id="S4.SS1.p2.6.m6.1.1.2.cmml" xref="S4.SS1.p2.6.m6.1.1.2">𝑖</ci><ci id="S4.SS1.p2.6.m6.1.1.3.cmml" xref="S4.SS1.p2.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_i < italic_j</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.9">For <math alttext="(i,j)" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.2"><semantics id="S4.SS1.p3.1.m1.2a"><mrow id="S4.SS1.p3.1.m1.2.3.2" xref="S4.SS1.p3.1.m1.2.3.1.cmml"><mo id="S4.SS1.p3.1.m1.2.3.2.1" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">i</mi><mo id="S4.SS1.p3.1.m1.2.3.2.2" xref="S4.SS1.p3.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS1.p3.1.m1.2.2" xref="S4.SS1.p3.1.m1.2.2.cmml">j</mi><mo id="S4.SS1.p3.1.m1.2.3.2.3" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.2b"><interval closure="open" id="S4.SS1.p3.1.m1.2.3.1.cmml" xref="S4.SS1.p3.1.m1.2.3.2"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝑖</ci><ci id="S4.SS1.p3.1.m1.2.2.cmml" xref="S4.SS1.p3.1.m1.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.2d">( italic_i , italic_j )</annotation></semantics></math> such that <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.2"><semantics id="S4.SS1.p3.2.m2.2a"><mrow id="S4.SS1.p3.2.m2.2.3" xref="S4.SS1.p3.2.m2.2.3.cmml"><msub id="S4.SS1.p3.2.m2.2.3.2" xref="S4.SS1.p3.2.m2.2.3.2.cmml"><mi id="S4.SS1.p3.2.m2.2.3.2.2" xref="S4.SS1.p3.2.m2.2.3.2.2.cmml">C</mi><mrow id="S4.SS1.p3.2.m2.2.2.2.4" xref="S4.SS1.p3.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p3.2.m2.1.1.1.1" xref="S4.SS1.p3.2.m2.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.2.m2.2.2.2.4.1" xref="S4.SS1.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.2.m2.2.2.2.2" xref="S4.SS1.p3.2.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p3.2.m2.2.3.1" xref="S4.SS1.p3.2.m2.2.3.1.cmml">=</mo><mn id="S4.SS1.p3.2.m2.2.3.3" xref="S4.SS1.p3.2.m2.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.2b"><apply id="S4.SS1.p3.2.m2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.3"><eq id="S4.SS1.p3.2.m2.2.3.1.cmml" xref="S4.SS1.p3.2.m2.2.3.1"></eq><apply id="S4.SS1.p3.2.m2.2.3.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.2.3.2.1.cmml" xref="S4.SS1.p3.2.m2.2.3.2">subscript</csymbol><ci id="S4.SS1.p3.2.m2.2.3.2.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2.2">𝐶</ci><list id="S4.SS1.p3.2.m2.2.2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.2.2.4"><ci id="S4.SS1.p3.2.m2.1.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.2.m2.2.2.2.2.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2">𝑗</ci></list></apply><cn id="S4.SS1.p3.2.m2.2.3.3.cmml" type="integer" xref="S4.SS1.p3.2.m2.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> (i.e., vertex <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">italic_i</annotation></semantics></math> and vertex <math alttext="j" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.1"><semantics id="S4.SS1.p3.4.m4.1a"><mi id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.1b"><ci id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">italic_j</annotation></semantics></math> are connected) we assign discrete coordinates of the edge <math alttext="x_{0}^{(a_{i,j})},...,x_{N_{i,j}}^{(a_{i,j})}" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.11"><semantics id="S4.SS1.p3.5.m5.11a"><mrow id="S4.SS1.p3.5.m5.11.11.2" xref="S4.SS1.p3.5.m5.11.11.3.cmml"><msubsup id="S4.SS1.p3.5.m5.10.10.1.1" xref="S4.SS1.p3.5.m5.10.10.1.1.cmml"><mi id="S4.SS1.p3.5.m5.10.10.1.1.2.2" xref="S4.SS1.p3.5.m5.10.10.1.1.2.2.cmml">x</mi><mn id="S4.SS1.p3.5.m5.10.10.1.1.2.3" xref="S4.SS1.p3.5.m5.10.10.1.1.2.3.cmml">0</mn><mrow id="S4.SS1.p3.5.m5.3.3.3.3" xref="S4.SS1.p3.5.m5.3.3.3.3.1.cmml"><mo id="S4.SS1.p3.5.m5.3.3.3.3.2" stretchy="false" xref="S4.SS1.p3.5.m5.3.3.3.3.1.cmml">(</mo><msub id="S4.SS1.p3.5.m5.3.3.3.3.1" xref="S4.SS1.p3.5.m5.3.3.3.3.1.cmml"><mi id="S4.SS1.p3.5.m5.3.3.3.3.1.2" xref="S4.SS1.p3.5.m5.3.3.3.3.1.2.cmml">a</mi><mrow id="S4.SS1.p3.5.m5.2.2.2.2.2.4" xref="S4.SS1.p3.5.m5.2.2.2.2.2.3.cmml"><mi id="S4.SS1.p3.5.m5.1.1.1.1.1.1" xref="S4.SS1.p3.5.m5.1.1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.5.m5.2.2.2.2.2.4.1" xref="S4.SS1.p3.5.m5.2.2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.5.m5.2.2.2.2.2.2" xref="S4.SS1.p3.5.m5.2.2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p3.5.m5.3.3.3.3.3" stretchy="false" xref="S4.SS1.p3.5.m5.3.3.3.3.1.cmml">)</mo></mrow></msubsup><mo id="S4.SS1.p3.5.m5.11.11.2.3" xref="S4.SS1.p3.5.m5.11.11.3.cmml">,</mo><mi id="S4.SS1.p3.5.m5.9.9" mathvariant="normal" xref="S4.SS1.p3.5.m5.9.9.cmml">…</mi><mo id="S4.SS1.p3.5.m5.11.11.2.4" xref="S4.SS1.p3.5.m5.11.11.3.cmml">,</mo><msubsup id="S4.SS1.p3.5.m5.11.11.2.2" xref="S4.SS1.p3.5.m5.11.11.2.2.cmml"><mi id="S4.SS1.p3.5.m5.11.11.2.2.2.2" xref="S4.SS1.p3.5.m5.11.11.2.2.2.2.cmml">x</mi><msub id="S4.SS1.p3.5.m5.5.5.2" xref="S4.SS1.p3.5.m5.5.5.2.cmml"><mi id="S4.SS1.p3.5.m5.5.5.2.4" xref="S4.SS1.p3.5.m5.5.5.2.4.cmml">N</mi><mrow id="S4.SS1.p3.5.m5.5.5.2.2.2.4" xref="S4.SS1.p3.5.m5.5.5.2.2.2.3.cmml"><mi id="S4.SS1.p3.5.m5.4.4.1.1.1.1" xref="S4.SS1.p3.5.m5.4.4.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.5.m5.5.5.2.2.2.4.1" xref="S4.SS1.p3.5.m5.5.5.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.5.m5.5.5.2.2.2.2" xref="S4.SS1.p3.5.m5.5.5.2.2.2.2.cmml">j</mi></mrow></msub><mrow id="S4.SS1.p3.5.m5.8.8.3.3" xref="S4.SS1.p3.5.m5.8.8.3.3.1.cmml"><mo id="S4.SS1.p3.5.m5.8.8.3.3.2" stretchy="false" xref="S4.SS1.p3.5.m5.8.8.3.3.1.cmml">(</mo><msub id="S4.SS1.p3.5.m5.8.8.3.3.1" xref="S4.SS1.p3.5.m5.8.8.3.3.1.cmml"><mi id="S4.SS1.p3.5.m5.8.8.3.3.1.2" xref="S4.SS1.p3.5.m5.8.8.3.3.1.2.cmml">a</mi><mrow id="S4.SS1.p3.5.m5.7.7.2.2.2.4" xref="S4.SS1.p3.5.m5.7.7.2.2.2.3.cmml"><mi id="S4.SS1.p3.5.m5.6.6.1.1.1.1" xref="S4.SS1.p3.5.m5.6.6.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.5.m5.7.7.2.2.2.4.1" xref="S4.SS1.p3.5.m5.7.7.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.5.m5.7.7.2.2.2.2" xref="S4.SS1.p3.5.m5.7.7.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p3.5.m5.8.8.3.3.3" stretchy="false" xref="S4.SS1.p3.5.m5.8.8.3.3.1.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.11b"><list id="S4.SS1.p3.5.m5.11.11.3.cmml" xref="S4.SS1.p3.5.m5.11.11.2"><apply id="S4.SS1.p3.5.m5.10.10.1.1.cmml" xref="S4.SS1.p3.5.m5.10.10.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.10.10.1.1.1.cmml" xref="S4.SS1.p3.5.m5.10.10.1.1">superscript</csymbol><apply id="S4.SS1.p3.5.m5.10.10.1.1.2.cmml" xref="S4.SS1.p3.5.m5.10.10.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.10.10.1.1.2.1.cmml" xref="S4.SS1.p3.5.m5.10.10.1.1">subscript</csymbol><ci id="S4.SS1.p3.5.m5.10.10.1.1.2.2.cmml" xref="S4.SS1.p3.5.m5.10.10.1.1.2.2">𝑥</ci><cn id="S4.SS1.p3.5.m5.10.10.1.1.2.3.cmml" type="integer" xref="S4.SS1.p3.5.m5.10.10.1.1.2.3">0</cn></apply><apply id="S4.SS1.p3.5.m5.3.3.3.3.1.cmml" xref="S4.SS1.p3.5.m5.3.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.3.3.3.3.1.1.cmml" xref="S4.SS1.p3.5.m5.3.3.3.3">subscript</csymbol><ci id="S4.SS1.p3.5.m5.3.3.3.3.1.2.cmml" xref="S4.SS1.p3.5.m5.3.3.3.3.1.2">𝑎</ci><list id="S4.SS1.p3.5.m5.2.2.2.2.2.3.cmml" xref="S4.SS1.p3.5.m5.2.2.2.2.2.4"><ci id="S4.SS1.p3.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.5.m5.2.2.2.2.2.2.cmml" xref="S4.SS1.p3.5.m5.2.2.2.2.2.2">𝑗</ci></list></apply></apply><ci id="S4.SS1.p3.5.m5.9.9.cmml" xref="S4.SS1.p3.5.m5.9.9">…</ci><apply id="S4.SS1.p3.5.m5.11.11.2.2.cmml" xref="S4.SS1.p3.5.m5.11.11.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.11.11.2.2.1.cmml" xref="S4.SS1.p3.5.m5.11.11.2.2">superscript</csymbol><apply id="S4.SS1.p3.5.m5.11.11.2.2.2.cmml" xref="S4.SS1.p3.5.m5.11.11.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.11.11.2.2.2.1.cmml" xref="S4.SS1.p3.5.m5.11.11.2.2">subscript</csymbol><ci id="S4.SS1.p3.5.m5.11.11.2.2.2.2.cmml" xref="S4.SS1.p3.5.m5.11.11.2.2.2.2">𝑥</ci><apply id="S4.SS1.p3.5.m5.5.5.2.cmml" xref="S4.SS1.p3.5.m5.5.5.2"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.5.5.2.3.cmml" xref="S4.SS1.p3.5.m5.5.5.2">subscript</csymbol><ci id="S4.SS1.p3.5.m5.5.5.2.4.cmml" xref="S4.SS1.p3.5.m5.5.5.2.4">𝑁</ci><list id="S4.SS1.p3.5.m5.5.5.2.2.2.3.cmml" xref="S4.SS1.p3.5.m5.5.5.2.2.2.4"><ci id="S4.SS1.p3.5.m5.4.4.1.1.1.1.cmml" xref="S4.SS1.p3.5.m5.4.4.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.5.m5.5.5.2.2.2.2.cmml" xref="S4.SS1.p3.5.m5.5.5.2.2.2.2">𝑗</ci></list></apply></apply><apply id="S4.SS1.p3.5.m5.8.8.3.3.1.cmml" xref="S4.SS1.p3.5.m5.8.8.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.8.8.3.3.1.1.cmml" xref="S4.SS1.p3.5.m5.8.8.3.3">subscript</csymbol><ci id="S4.SS1.p3.5.m5.8.8.3.3.1.2.cmml" xref="S4.SS1.p3.5.m5.8.8.3.3.1.2">𝑎</ci><list id="S4.SS1.p3.5.m5.7.7.2.2.2.3.cmml" xref="S4.SS1.p3.5.m5.7.7.2.2.2.4"><ci id="S4.SS1.p3.5.m5.6.6.1.1.1.1.cmml" xref="S4.SS1.p3.5.m5.6.6.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.5.m5.7.7.2.2.2.2.cmml" xref="S4.SS1.p3.5.m5.7.7.2.2.2.2">𝑗</ci></list></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.11c">x_{0}^{(a_{i,j})},...,x_{N_{i,j}}^{(a_{i,j})}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.11d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math>, which has step size <math alttext="a_{i,j}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.2"><semantics id="S4.SS1.p3.6.m6.2a"><msub id="S4.SS1.p3.6.m6.2.3" xref="S4.SS1.p3.6.m6.2.3.cmml"><mi id="S4.SS1.p3.6.m6.2.3.2" xref="S4.SS1.p3.6.m6.2.3.2.cmml">a</mi><mrow id="S4.SS1.p3.6.m6.2.2.2.4" xref="S4.SS1.p3.6.m6.2.2.2.3.cmml"><mi id="S4.SS1.p3.6.m6.1.1.1.1" xref="S4.SS1.p3.6.m6.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.6.m6.2.2.2.4.1" xref="S4.SS1.p3.6.m6.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.6.m6.2.2.2.2" xref="S4.SS1.p3.6.m6.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.2b"><apply id="S4.SS1.p3.6.m6.2.3.cmml" xref="S4.SS1.p3.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m6.2.3.1.cmml" xref="S4.SS1.p3.6.m6.2.3">subscript</csymbol><ci id="S4.SS1.p3.6.m6.2.3.2.cmml" xref="S4.SS1.p3.6.m6.2.3.2">𝑎</ci><list id="S4.SS1.p3.6.m6.2.2.2.3.cmml" xref="S4.SS1.p3.6.m6.2.2.2.4"><ci id="S4.SS1.p3.6.m6.1.1.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.6.m6.2.2.2.2.cmml" xref="S4.SS1.p3.6.m6.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.2c">a_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.2d">italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Here <math alttext="x_{0}^{(a_{i,j})}=0" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m7.3"><semantics id="S4.SS1.p3.7.m7.3a"><mrow id="S4.SS1.p3.7.m7.3.4" xref="S4.SS1.p3.7.m7.3.4.cmml"><msubsup id="S4.SS1.p3.7.m7.3.4.2" xref="S4.SS1.p3.7.m7.3.4.2.cmml"><mi id="S4.SS1.p3.7.m7.3.4.2.2.2" xref="S4.SS1.p3.7.m7.3.4.2.2.2.cmml">x</mi><mn id="S4.SS1.p3.7.m7.3.4.2.2.3" xref="S4.SS1.p3.7.m7.3.4.2.2.3.cmml">0</mn><mrow id="S4.SS1.p3.7.m7.3.3.3.3" xref="S4.SS1.p3.7.m7.3.3.3.3.1.cmml"><mo id="S4.SS1.p3.7.m7.3.3.3.3.2" stretchy="false" xref="S4.SS1.p3.7.m7.3.3.3.3.1.cmml">(</mo><msub id="S4.SS1.p3.7.m7.3.3.3.3.1" xref="S4.SS1.p3.7.m7.3.3.3.3.1.cmml"><mi id="S4.SS1.p3.7.m7.3.3.3.3.1.2" xref="S4.SS1.p3.7.m7.3.3.3.3.1.2.cmml">a</mi><mrow id="S4.SS1.p3.7.m7.2.2.2.2.2.4" xref="S4.SS1.p3.7.m7.2.2.2.2.2.3.cmml"><mi id="S4.SS1.p3.7.m7.1.1.1.1.1.1" xref="S4.SS1.p3.7.m7.1.1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.7.m7.2.2.2.2.2.4.1" xref="S4.SS1.p3.7.m7.2.2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.7.m7.2.2.2.2.2.2" xref="S4.SS1.p3.7.m7.2.2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p3.7.m7.3.3.3.3.3" stretchy="false" xref="S4.SS1.p3.7.m7.3.3.3.3.1.cmml">)</mo></mrow></msubsup><mo id="S4.SS1.p3.7.m7.3.4.1" xref="S4.SS1.p3.7.m7.3.4.1.cmml">=</mo><mn id="S4.SS1.p3.7.m7.3.4.3" xref="S4.SS1.p3.7.m7.3.4.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m7.3b"><apply id="S4.SS1.p3.7.m7.3.4.cmml" xref="S4.SS1.p3.7.m7.3.4"><eq id="S4.SS1.p3.7.m7.3.4.1.cmml" xref="S4.SS1.p3.7.m7.3.4.1"></eq><apply id="S4.SS1.p3.7.m7.3.4.2.cmml" xref="S4.SS1.p3.7.m7.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m7.3.4.2.1.cmml" xref="S4.SS1.p3.7.m7.3.4.2">superscript</csymbol><apply id="S4.SS1.p3.7.m7.3.4.2.2.cmml" xref="S4.SS1.p3.7.m7.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m7.3.4.2.2.1.cmml" xref="S4.SS1.p3.7.m7.3.4.2">subscript</csymbol><ci id="S4.SS1.p3.7.m7.3.4.2.2.2.cmml" xref="S4.SS1.p3.7.m7.3.4.2.2.2">𝑥</ci><cn id="S4.SS1.p3.7.m7.3.4.2.2.3.cmml" type="integer" xref="S4.SS1.p3.7.m7.3.4.2.2.3">0</cn></apply><apply id="S4.SS1.p3.7.m7.3.3.3.3.1.cmml" xref="S4.SS1.p3.7.m7.3.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m7.3.3.3.3.1.1.cmml" xref="S4.SS1.p3.7.m7.3.3.3.3">subscript</csymbol><ci id="S4.SS1.p3.7.m7.3.3.3.3.1.2.cmml" xref="S4.SS1.p3.7.m7.3.3.3.3.1.2">𝑎</ci><list id="S4.SS1.p3.7.m7.2.2.2.2.2.3.cmml" xref="S4.SS1.p3.7.m7.2.2.2.2.2.4"><ci id="S4.SS1.p3.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.7.m7.2.2.2.2.2.2.cmml" xref="S4.SS1.p3.7.m7.2.2.2.2.2.2">𝑗</ci></list></apply></apply><cn id="S4.SS1.p3.7.m7.3.4.3.cmml" type="integer" xref="S4.SS1.p3.7.m7.3.4.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m7.3c">x_{0}^{(a_{i,j})}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m7.3d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT = 0</annotation></semantics></math> and <math alttext="x_{N_{i,j}}^{a_{i,j}}" class="ltx_Math" display="inline" id="S4.SS1.p3.8.m8.4"><semantics id="S4.SS1.p3.8.m8.4a"><msubsup id="S4.SS1.p3.8.m8.4.5" xref="S4.SS1.p3.8.m8.4.5.cmml"><mi id="S4.SS1.p3.8.m8.4.5.2.2" xref="S4.SS1.p3.8.m8.4.5.2.2.cmml">x</mi><msub id="S4.SS1.p3.8.m8.2.2.2" xref="S4.SS1.p3.8.m8.2.2.2.cmml"><mi id="S4.SS1.p3.8.m8.2.2.2.4" xref="S4.SS1.p3.8.m8.2.2.2.4.cmml">N</mi><mrow id="S4.SS1.p3.8.m8.2.2.2.2.2.4" xref="S4.SS1.p3.8.m8.2.2.2.2.2.3.cmml"><mi id="S4.SS1.p3.8.m8.1.1.1.1.1.1" xref="S4.SS1.p3.8.m8.1.1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.8.m8.2.2.2.2.2.4.1" xref="S4.SS1.p3.8.m8.2.2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.8.m8.2.2.2.2.2.2" xref="S4.SS1.p3.8.m8.2.2.2.2.2.2.cmml">j</mi></mrow></msub><msub id="S4.SS1.p3.8.m8.4.4.2" xref="S4.SS1.p3.8.m8.4.4.2.cmml"><mi id="S4.SS1.p3.8.m8.4.4.2.4" xref="S4.SS1.p3.8.m8.4.4.2.4.cmml">a</mi><mrow id="S4.SS1.p3.8.m8.4.4.2.2.2.4" xref="S4.SS1.p3.8.m8.4.4.2.2.2.3.cmml"><mi id="S4.SS1.p3.8.m8.3.3.1.1.1.1" xref="S4.SS1.p3.8.m8.3.3.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.8.m8.4.4.2.2.2.4.1" xref="S4.SS1.p3.8.m8.4.4.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.8.m8.4.4.2.2.2.2" xref="S4.SS1.p3.8.m8.4.4.2.2.2.2.cmml">j</mi></mrow></msub></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.8.m8.4b"><apply id="S4.SS1.p3.8.m8.4.5.cmml" xref="S4.SS1.p3.8.m8.4.5"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.4.5.1.cmml" xref="S4.SS1.p3.8.m8.4.5">superscript</csymbol><apply id="S4.SS1.p3.8.m8.4.5.2.cmml" xref="S4.SS1.p3.8.m8.4.5"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.4.5.2.1.cmml" xref="S4.SS1.p3.8.m8.4.5">subscript</csymbol><ci id="S4.SS1.p3.8.m8.4.5.2.2.cmml" xref="S4.SS1.p3.8.m8.4.5.2.2">𝑥</ci><apply id="S4.SS1.p3.8.m8.2.2.2.cmml" xref="S4.SS1.p3.8.m8.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.2.2.2.3.cmml" xref="S4.SS1.p3.8.m8.2.2.2">subscript</csymbol><ci id="S4.SS1.p3.8.m8.2.2.2.4.cmml" xref="S4.SS1.p3.8.m8.2.2.2.4">𝑁</ci><list id="S4.SS1.p3.8.m8.2.2.2.2.2.3.cmml" xref="S4.SS1.p3.8.m8.2.2.2.2.2.4"><ci id="S4.SS1.p3.8.m8.1.1.1.1.1.1.cmml" xref="S4.SS1.p3.8.m8.1.1.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.8.m8.2.2.2.2.2.2.cmml" xref="S4.SS1.p3.8.m8.2.2.2.2.2.2">𝑗</ci></list></apply></apply><apply id="S4.SS1.p3.8.m8.4.4.2.cmml" xref="S4.SS1.p3.8.m8.4.4.2"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.4.4.2.3.cmml" xref="S4.SS1.p3.8.m8.4.4.2">subscript</csymbol><ci id="S4.SS1.p3.8.m8.4.4.2.4.cmml" xref="S4.SS1.p3.8.m8.4.4.2.4">𝑎</ci><list id="S4.SS1.p3.8.m8.4.4.2.2.2.3.cmml" xref="S4.SS1.p3.8.m8.4.4.2.2.2.4"><ci id="S4.SS1.p3.8.m8.3.3.1.1.1.1.cmml" xref="S4.SS1.p3.8.m8.3.3.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.8.m8.4.4.2.2.2.2.cmml" xref="S4.SS1.p3.8.m8.4.4.2.2.2.2">𝑗</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.8.m8.4c">x_{N_{i,j}}^{a_{i,j}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.8.m8.4d">italic_x start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is the right endpoint. In other words, each edge of the graph is an interval with discrete points starting at zero and ending at <math alttext="x_{N_{i,j}}^{(a_{i,j})}=L_{i,j}=N_{i,j}a_{i,j}" class="ltx_Math" display="inline" id="S4.SS1.p3.9.m9.11"><semantics id="S4.SS1.p3.9.m9.11a"><mrow id="S4.SS1.p3.9.m9.11.12" xref="S4.SS1.p3.9.m9.11.12.cmml"><msubsup id="S4.SS1.p3.9.m9.11.12.2" xref="S4.SS1.p3.9.m9.11.12.2.cmml"><mi id="S4.SS1.p3.9.m9.11.12.2.2.2" xref="S4.SS1.p3.9.m9.11.12.2.2.2.cmml">x</mi><msub id="S4.SS1.p3.9.m9.2.2.2" xref="S4.SS1.p3.9.m9.2.2.2.cmml"><mi id="S4.SS1.p3.9.m9.2.2.2.4" xref="S4.SS1.p3.9.m9.2.2.2.4.cmml">N</mi><mrow id="S4.SS1.p3.9.m9.2.2.2.2.2.4" xref="S4.SS1.p3.9.m9.2.2.2.2.2.3.cmml"><mi id="S4.SS1.p3.9.m9.1.1.1.1.1.1" xref="S4.SS1.p3.9.m9.1.1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.9.m9.2.2.2.2.2.4.1" xref="S4.SS1.p3.9.m9.2.2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.9.m9.2.2.2.2.2.2" xref="S4.SS1.p3.9.m9.2.2.2.2.2.2.cmml">j</mi></mrow></msub><mrow id="S4.SS1.p3.9.m9.5.5.3.3" 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end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.4">The wavefunction <math alttext="\Psi" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mi id="S4.SS1.p4.1.m1.1.1" mathvariant="normal" xref="S4.SS1.p4.1.m1.1.1.cmml">Ψ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><ci id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1">Ψ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">\Psi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">roman_Ψ</annotation></semantics></math> is a vector with <math alttext="E" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mi id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><ci id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">italic_E</annotation></semantics></math> components and each of its components is a vector in <math alttext="\mathbb{C}^{N_{i,j}+1}" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m3.2"><semantics id="S4.SS1.p4.3.m3.2a"><msup id="S4.SS1.p4.3.m3.2.3" xref="S4.SS1.p4.3.m3.2.3.cmml"><mi id="S4.SS1.p4.3.m3.2.3.2" xref="S4.SS1.p4.3.m3.2.3.2.cmml">ℂ</mi><mrow id="S4.SS1.p4.3.m3.2.2.2" xref="S4.SS1.p4.3.m3.2.2.2.cmml"><msub id="S4.SS1.p4.3.m3.2.2.2.4" xref="S4.SS1.p4.3.m3.2.2.2.4.cmml"><mi id="S4.SS1.p4.3.m3.2.2.2.4.2" xref="S4.SS1.p4.3.m3.2.2.2.4.2.cmml">N</mi><mrow id="S4.SS1.p4.3.m3.2.2.2.2.2.4" xref="S4.SS1.p4.3.m3.2.2.2.2.2.3.cmml"><mi id="S4.SS1.p4.3.m3.1.1.1.1.1.1" xref="S4.SS1.p4.3.m3.1.1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p4.3.m3.2.2.2.2.2.4.1" 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start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) ) start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.6">with <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m1.2"><semantics id="S4.SS1.p4.5.m1.2a"><mrow id="S4.SS1.p4.5.m1.2.3" xref="S4.SS1.p4.5.m1.2.3.cmml"><msub id="S4.SS1.p4.5.m1.2.3.2" xref="S4.SS1.p4.5.m1.2.3.2.cmml"><mi id="S4.SS1.p4.5.m1.2.3.2.2" xref="S4.SS1.p4.5.m1.2.3.2.2.cmml">C</mi><mrow id="S4.SS1.p4.5.m1.2.2.2.4" xref="S4.SS1.p4.5.m1.2.2.2.3.cmml"><mi id="S4.SS1.p4.5.m1.1.1.1.1" 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xref="S4.SS1.p4.5.m1.2.2.2.2">𝑗</ci></list></apply><cn id="S4.SS1.p4.5.m1.2.3.3.cmml" type="integer" xref="S4.SS1.p4.5.m1.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m1.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m1.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>, where <math alttext="i<j" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m2.1"><semantics id="S4.SS1.p4.6.m2.1a"><mrow id="S4.SS1.p4.6.m2.1.1" xref="S4.SS1.p4.6.m2.1.1.cmml"><mi id="S4.SS1.p4.6.m2.1.1.2" xref="S4.SS1.p4.6.m2.1.1.2.cmml">i</mi><mo id="S4.SS1.p4.6.m2.1.1.1" xref="S4.SS1.p4.6.m2.1.1.1.cmml"><</mo><mi id="S4.SS1.p4.6.m2.1.1.3" xref="S4.SS1.p4.6.m2.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.6.m2.1b"><apply id="S4.SS1.p4.6.m2.1.1.cmml" xref="S4.SS1.p4.6.m2.1.1"><lt id="S4.SS1.p4.6.m2.1.1.1.cmml" 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id="S4.E28.m1.25.25.25.2.2.2" mathvariant="normal" xref="S4.E28.m1.25.25.25.2.2.2.cmml">Ψ</mi><mrow id="S4.E28.m1.26.26.26.3.3.3.1.4" xref="S4.E28.m1.26.26.26.3.3.3.1.3.cmml"><mi id="S4.E28.m1.26.26.26.3.3.3.1.1" xref="S4.E28.m1.26.26.26.3.3.3.1.1.cmml">i</mi><mo id="S4.E28.m1.26.26.26.3.3.3.1.4.1" xref="S4.E28.m1.26.26.26.3.3.3.1.3.cmml">,</mo><mi id="S4.E28.m1.26.26.26.3.3.3.1.2" xref="S4.E28.m1.26.26.26.3.3.3.1.2.cmml">j</mi></mrow><mrow id="S4.E28.m1.27.27.27.4.4.4.1.3" xref="S4.E28.m1.27.27.27.4.4.4.1.3.1.cmml"><mo id="S4.E28.m1.27.27.27.4.4.4.1.3.2" stretchy="false" xref="S4.E28.m1.27.27.27.4.4.4.1.3.1.cmml">(</mo><msub id="S4.E28.m1.27.27.27.4.4.4.1.3.1" xref="S4.E28.m1.27.27.27.4.4.4.1.3.1.cmml"><mi id="S4.E28.m1.27.27.27.4.4.4.1.3.1.2" xref="S4.E28.m1.27.27.27.4.4.4.1.3.1.2.cmml">a</mi><mrow id="S4.E28.m1.27.27.27.4.4.4.1.2.2.4" xref="S4.E28.m1.27.27.27.4.4.4.1.2.2.3.cmml"><mi id="S4.E28.m1.27.27.27.4.4.4.1.1.1.1" xref="S4.E28.m1.27.27.27.4.4.4.1.1.1.1.cmml">i</mi><mo 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id="S4.E28.m1.46.46.46.23.23.23.1.1.1.1.cmml" xref="S4.E28.m1.46.46.46.23.23.23.1.1.1.1">𝑖</ci><ci id="S4.E28.m1.46.46.46.23.23.23.1.2.2.2.cmml" xref="S4.E28.m1.46.46.46.23.23.23.1.2.2.2">𝑗</ci></list></apply></apply></apply></apply><cn id="S4.E28.m1.49.49.49.26.26.26.cmml" type="integer" xref="S4.E28.m1.49.49.49.26.26.26">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E28.m1.51c">\displaystyle\begin{split}&\frac{1}{a_{i,j}^{2}}[\Psi_{i,j}^{(a_{i,j})}(x_{n_{% i,j}}^{(a_{i,j})}-a_{i,j})-2\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})\\ &+\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})}+a_{i,j})]+k^{2}\Psi_{i,j}^{(a% _{i,j})}(x_{n_{i,j}}^{(a_{i,j})})=0,\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E28.m1.51d">start_ROW start_CELL end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT - italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) - 2 roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ] + italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) = 0 , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(28)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p5.3">where <math alttext="1\leq n_{i.j}\leq N_{i,j}-1" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.4"><semantics id="S4.SS1.p5.1.m1.4a"><mrow id="S4.SS1.p5.1.m1.4.5" xref="S4.SS1.p5.1.m1.4.5.cmml"><mn id="S4.SS1.p5.1.m1.4.5.2" xref="S4.SS1.p5.1.m1.4.5.2.cmml">1</mn><mo id="S4.SS1.p5.1.m1.4.5.3" xref="S4.SS1.p5.1.m1.4.5.3.cmml">≤</mo><msub id="S4.SS1.p5.1.m1.4.5.4" xref="S4.SS1.p5.1.m1.4.5.4.cmml"><mi id="S4.SS1.p5.1.m1.4.5.4.2" xref="S4.SS1.p5.1.m1.4.5.4.2.cmml">n</mi><mrow id="S4.SS1.p5.1.m1.2.2.2.4" xref="S4.SS1.p5.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p5.1.m1.1.1.1.1" xref="S4.SS1.p5.1.m1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p5.1.m1.2.2.2.4.1" lspace="0em" rspace="0.167em" xref="S4.SS1.p5.1.m1.2.2.2.3a.cmml">.</mo><mi id="S4.SS1.p5.1.m1.2.2.2.2" xref="S4.SS1.p5.1.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p5.1.m1.4.5.5" xref="S4.SS1.p5.1.m1.4.5.5.cmml">≤</mo><mrow id="S4.SS1.p5.1.m1.4.5.6" xref="S4.SS1.p5.1.m1.4.5.6.cmml"><msub id="S4.SS1.p5.1.m1.4.5.6.2" xref="S4.SS1.p5.1.m1.4.5.6.2.cmml"><mi id="S4.SS1.p5.1.m1.4.5.6.2.2" xref="S4.SS1.p5.1.m1.4.5.6.2.2.cmml">N</mi><mrow id="S4.SS1.p5.1.m1.4.4.2.4" xref="S4.SS1.p5.1.m1.4.4.2.3.cmml"><mi id="S4.SS1.p5.1.m1.3.3.1.1" xref="S4.SS1.p5.1.m1.3.3.1.1.cmml">i</mi><mo id="S4.SS1.p5.1.m1.4.4.2.4.1" xref="S4.SS1.p5.1.m1.4.4.2.3.cmml">,</mo><mi id="S4.SS1.p5.1.m1.4.4.2.2" xref="S4.SS1.p5.1.m1.4.4.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p5.1.m1.4.5.6.1" xref="S4.SS1.p5.1.m1.4.5.6.1.cmml">−</mo><mn id="S4.SS1.p5.1.m1.4.5.6.3" xref="S4.SS1.p5.1.m1.4.5.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.4b"><apply id="S4.SS1.p5.1.m1.4.5.cmml" xref="S4.SS1.p5.1.m1.4.5"><and id="S4.SS1.p5.1.m1.4.5a.cmml" xref="S4.SS1.p5.1.m1.4.5"></and><apply 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href="https://arxiv.org/html/2411.14397v1#S4.SS1.p5.1.m1.4.5.4.cmml" id="S4.SS1.p5.1.m1.4.5d.cmml" xref="S4.SS1.p5.1.m1.4.5"></share><apply id="S4.SS1.p5.1.m1.4.5.6.cmml" xref="S4.SS1.p5.1.m1.4.5.6"><minus id="S4.SS1.p5.1.m1.4.5.6.1.cmml" xref="S4.SS1.p5.1.m1.4.5.6.1"></minus><apply id="S4.SS1.p5.1.m1.4.5.6.2.cmml" xref="S4.SS1.p5.1.m1.4.5.6.2"><csymbol cd="ambiguous" id="S4.SS1.p5.1.m1.4.5.6.2.1.cmml" xref="S4.SS1.p5.1.m1.4.5.6.2">subscript</csymbol><ci id="S4.SS1.p5.1.m1.4.5.6.2.2.cmml" xref="S4.SS1.p5.1.m1.4.5.6.2.2">𝑁</ci><list id="S4.SS1.p5.1.m1.4.4.2.3.cmml" xref="S4.SS1.p5.1.m1.4.4.2.4"><ci id="S4.SS1.p5.1.m1.3.3.1.1.cmml" xref="S4.SS1.p5.1.m1.3.3.1.1">𝑖</ci><ci id="S4.SS1.p5.1.m1.4.4.2.2.cmml" xref="S4.SS1.p5.1.m1.4.4.2.2">𝑗</ci></list></apply><cn id="S4.SS1.p5.1.m1.4.5.6.3.cmml" type="integer" xref="S4.SS1.p5.1.m1.4.5.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.4c">1\leq n_{i.j}\leq N_{i,j}-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.4d">1 ≤ italic_n start_POSTSUBSCRIPT italic_i . italic_j end_POSTSUBSCRIPT ≤ italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - 1</annotation></semantics></math> and <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m2.2"><semantics id="S4.SS1.p5.2.m2.2a"><mrow id="S4.SS1.p5.2.m2.2.3" xref="S4.SS1.p5.2.m2.2.3.cmml"><msub id="S4.SS1.p5.2.m2.2.3.2" xref="S4.SS1.p5.2.m2.2.3.2.cmml"><mi id="S4.SS1.p5.2.m2.2.3.2.2" xref="S4.SS1.p5.2.m2.2.3.2.2.cmml">C</mi><mrow id="S4.SS1.p5.2.m2.2.2.2.4" xref="S4.SS1.p5.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p5.2.m2.1.1.1.1" xref="S4.SS1.p5.2.m2.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p5.2.m2.2.2.2.4.1" xref="S4.SS1.p5.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p5.2.m2.2.2.2.2" xref="S4.SS1.p5.2.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p5.2.m2.2.3.1" xref="S4.SS1.p5.2.m2.2.3.1.cmml">=</mo><mn id="S4.SS1.p5.2.m2.2.3.3" 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start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>, <math alttext="i<j" class="ltx_Math" display="inline" id="S4.SS1.p5.3.m3.1"><semantics id="S4.SS1.p5.3.m3.1a"><mrow id="S4.SS1.p5.3.m3.1.1" xref="S4.SS1.p5.3.m3.1.1.cmml"><mi id="S4.SS1.p5.3.m3.1.1.2" xref="S4.SS1.p5.3.m3.1.1.2.cmml">i</mi><mo id="S4.SS1.p5.3.m3.1.1.1" xref="S4.SS1.p5.3.m3.1.1.1.cmml"><</mo><mi id="S4.SS1.p5.3.m3.1.1.3" xref="S4.SS1.p5.3.m3.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.3.m3.1b"><apply id="S4.SS1.p5.3.m3.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1"><lt id="S4.SS1.p5.3.m3.1.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1.1"></lt><ci id="S4.SS1.p5.3.m3.1.1.2.cmml" xref="S4.SS1.p5.3.m3.1.1.2">𝑖</ci><ci id="S4.SS1.p5.3.m3.1.1.3.cmml" xref="S4.SS1.p5.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.3.m3.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.3.m3.1d">italic_i < italic_j</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p6"> <p class="ltx_p" id="S4.SS1.p6.6">We use the standard Euclidean inner product on the discrete graph</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx22"> <tbody id="S4.E29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\langle\Psi,\Phi\rangle=\sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{V}\sum_{n_{i,j}=0}^{N_{i,j}}C_{i,j}\Psi_{i,j}^{(a_{i,j})}(% x_{n_{i,j}}^{(a_{i,j})})\Phi_{i,j}^{(a_{i,j})*}(x_{n_{i,j}}^{(a_{i,j})})." class="ltx_Math" display="inline" id="S4.E29.m1.31"><semantics id="S4.E29.m1.31a"><mrow id="S4.E29.m1.31.31.1" xref="S4.E29.m1.31.31.1.1.cmml"><mrow id="S4.E29.m1.31.31.1.1" xref="S4.E29.m1.31.31.1.1.cmml"><mrow id="S4.E29.m1.31.31.1.1.4.2" xref="S4.E29.m1.31.31.1.1.4.1.cmml"><mo 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xref="S4.E29.m1.26.26.1.1.1.1">𝑖</ci><ci id="S4.E29.m1.27.27.2.2.2.2.cmml" xref="S4.E29.m1.27.27.2.2.2.2">𝑗</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E29.m1.31c">\displaystyle\langle\Psi,\Phi\rangle=\sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{V}\sum_{n_{i,j}=0}^{N_{i,j}}C_{i,j}\Psi_{i,j}^{(a_{i,j})}(% x_{n_{i,j}}^{(a_{i,j})})\Phi_{i,j}^{(a_{i,j})*}(x_{n_{i,j}}^{(a_{i,j})}).</annotation><annotation encoding="application/x-llamapun" id="S4.E29.m1.31d">⟨ roman_Ψ , roman_Φ ⟩ = ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_i , italic_j = 1 end_CELL end_ROW start_ROW start_CELL italic_i < italic_j end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) roman_Φ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ∗ end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(29)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p6.2">where elements of <math alttext="E" class="ltx_Math" display="inline" id="S4.SS1.p6.1.m1.1"><semantics id="S4.SS1.p6.1.m1.1a"><mi id="S4.SS1.p6.1.m1.1.1" xref="S4.SS1.p6.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.1.m1.1b"><ci id="S4.SS1.p6.1.m1.1.1.cmml" xref="S4.SS1.p6.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.1.m1.1d">italic_E</annotation></semantics></math> component wavefunction vector <math alttext="\Phi" class="ltx_Math" display="inline" id="S4.SS1.p6.2.m2.1"><semantics id="S4.SS1.p6.2.m2.1a"><mi id="S4.SS1.p6.2.m2.1.1" mathvariant="normal" xref="S4.SS1.p6.2.m2.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.2.m2.1b"><ci id="S4.SS1.p6.2.m2.1.1.cmml" xref="S4.SS1.p6.2.m2.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.2.m2.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.2.m2.1d">roman_Φ</annotation></semantics></math> on the discrete graph are defined as</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\left(\Phi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})\right)_{n_{i,j}=0}^{N_{i% ,j}}," class="ltx_Math" display="block" id="S4.Ex14.m1.15"><semantics id="S4.Ex14.m1.15a"><mrow id="S4.Ex14.m1.15.15.1" xref="S4.Ex14.m1.15.15.1.1.cmml"><msubsup id="S4.Ex14.m1.15.15.1.1" xref="S4.Ex14.m1.15.15.1.1.cmml"><mrow id="S4.Ex14.m1.15.15.1.1.1.1.1" 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id="S4.Ex14.m1.12.12.2.4.cmml" xref="S4.Ex14.m1.12.12.2.4"><csymbol cd="ambiguous" id="S4.Ex14.m1.12.12.2.4.1.cmml" xref="S4.Ex14.m1.12.12.2.4">subscript</csymbol><ci id="S4.Ex14.m1.12.12.2.4.2.cmml" xref="S4.Ex14.m1.12.12.2.4.2">𝑛</ci><list id="S4.Ex14.m1.12.12.2.2.2.3.cmml" xref="S4.Ex14.m1.12.12.2.2.2.4"><ci id="S4.Ex14.m1.11.11.1.1.1.1.cmml" xref="S4.Ex14.m1.11.11.1.1.1.1">𝑖</ci><ci id="S4.Ex14.m1.12.12.2.2.2.2.cmml" xref="S4.Ex14.m1.12.12.2.2.2.2">𝑗</ci></list></apply><cn id="S4.Ex14.m1.12.12.2.5.cmml" type="integer" xref="S4.Ex14.m1.12.12.2.5">0</cn></apply></apply><apply id="S4.Ex14.m1.14.14.2.cmml" xref="S4.Ex14.m1.14.14.2"><csymbol cd="ambiguous" id="S4.Ex14.m1.14.14.2.3.cmml" xref="S4.Ex14.m1.14.14.2">subscript</csymbol><ci id="S4.Ex14.m1.14.14.2.4.cmml" xref="S4.Ex14.m1.14.14.2.4">𝑁</ci><list id="S4.Ex14.m1.14.14.2.2.2.3.cmml" xref="S4.Ex14.m1.14.14.2.2.2.4"><ci id="S4.Ex14.m1.13.13.1.1.1.1.cmml" xref="S4.Ex14.m1.13.13.1.1.1.1">𝑖</ci><ci id="S4.Ex14.m1.14.14.2.2.2.2.cmml" xref="S4.Ex14.m1.14.14.2.2.2.2">𝑗</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m1.15c">\left(\Phi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})\right)_{n_{i,j}=0}^{N_{i% ,j}},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.15d">( roman_Φ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) ) start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p6.5">for <math alttext="(i,j)" class="ltx_Math" display="inline" id="S4.SS1.p6.3.m1.2"><semantics id="S4.SS1.p6.3.m1.2a"><mrow id="S4.SS1.p6.3.m1.2.3.2" xref="S4.SS1.p6.3.m1.2.3.1.cmml"><mo id="S4.SS1.p6.3.m1.2.3.2.1" stretchy="false" xref="S4.SS1.p6.3.m1.2.3.1.cmml">(</mo><mi id="S4.SS1.p6.3.m1.1.1" xref="S4.SS1.p6.3.m1.1.1.cmml">i</mi><mo id="S4.SS1.p6.3.m1.2.3.2.2" xref="S4.SS1.p6.3.m1.2.3.1.cmml">,</mo><mi id="S4.SS1.p6.3.m1.2.2" xref="S4.SS1.p6.3.m1.2.2.cmml">j</mi><mo id="S4.SS1.p6.3.m1.2.3.2.3" stretchy="false" xref="S4.SS1.p6.3.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.3.m1.2b"><interval closure="open" id="S4.SS1.p6.3.m1.2.3.1.cmml" xref="S4.SS1.p6.3.m1.2.3.2"><ci id="S4.SS1.p6.3.m1.1.1.cmml" xref="S4.SS1.p6.3.m1.1.1">𝑖</ci><ci id="S4.SS1.p6.3.m1.2.2.cmml" xref="S4.SS1.p6.3.m1.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.3.m1.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.3.m1.2d">( italic_i , italic_j )</annotation></semantics></math> such that <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S4.SS1.p6.4.m2.2"><semantics id="S4.SS1.p6.4.m2.2a"><mrow id="S4.SS1.p6.4.m2.2.3" xref="S4.SS1.p6.4.m2.2.3.cmml"><msub id="S4.SS1.p6.4.m2.2.3.2" xref="S4.SS1.p6.4.m2.2.3.2.cmml"><mi id="S4.SS1.p6.4.m2.2.3.2.2" xref="S4.SS1.p6.4.m2.2.3.2.2.cmml">C</mi><mrow id="S4.SS1.p6.4.m2.2.2.2.4" xref="S4.SS1.p6.4.m2.2.2.2.3.cmml"><mi id="S4.SS1.p6.4.m2.1.1.1.1" xref="S4.SS1.p6.4.m2.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p6.4.m2.2.2.2.4.1" xref="S4.SS1.p6.4.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p6.4.m2.2.2.2.2" xref="S4.SS1.p6.4.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p6.4.m2.2.3.1" xref="S4.SS1.p6.4.m2.2.3.1.cmml">=</mo><mn id="S4.SS1.p6.4.m2.2.3.3" xref="S4.SS1.p6.4.m2.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.4.m2.2b"><apply id="S4.SS1.p6.4.m2.2.3.cmml" xref="S4.SS1.p6.4.m2.2.3"><eq id="S4.SS1.p6.4.m2.2.3.1.cmml" xref="S4.SS1.p6.4.m2.2.3.1"></eq><apply id="S4.SS1.p6.4.m2.2.3.2.cmml" xref="S4.SS1.p6.4.m2.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p6.4.m2.2.3.2.1.cmml" xref="S4.SS1.p6.4.m2.2.3.2">subscript</csymbol><ci id="S4.SS1.p6.4.m2.2.3.2.2.cmml" xref="S4.SS1.p6.4.m2.2.3.2.2">𝐶</ci><list id="S4.SS1.p6.4.m2.2.2.2.3.cmml" xref="S4.SS1.p6.4.m2.2.2.2.4"><ci id="S4.SS1.p6.4.m2.1.1.1.1.cmml" xref="S4.SS1.p6.4.m2.1.1.1.1">𝑖</ci><ci id="S4.SS1.p6.4.m2.2.2.2.2.cmml" xref="S4.SS1.p6.4.m2.2.2.2.2">𝑗</ci></list></apply><cn id="S4.SS1.p6.4.m2.2.3.3.cmml" type="integer" xref="S4.SS1.p6.4.m2.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.4.m2.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.4.m2.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>, where <math alttext="i<j" class="ltx_Math" display="inline" id="S4.SS1.p6.5.m3.1"><semantics id="S4.SS1.p6.5.m3.1a"><mrow id="S4.SS1.p6.5.m3.1.1" xref="S4.SS1.p6.5.m3.1.1.cmml"><mi id="S4.SS1.p6.5.m3.1.1.2" xref="S4.SS1.p6.5.m3.1.1.2.cmml">i</mi><mo id="S4.SS1.p6.5.m3.1.1.1" xref="S4.SS1.p6.5.m3.1.1.1.cmml"><</mo><mi id="S4.SS1.p6.5.m3.1.1.3" xref="S4.SS1.p6.5.m3.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.5.m3.1b"><apply id="S4.SS1.p6.5.m3.1.1.cmml" xref="S4.SS1.p6.5.m3.1.1"><lt id="S4.SS1.p6.5.m3.1.1.1.cmml" xref="S4.SS1.p6.5.m3.1.1.1"></lt><ci id="S4.SS1.p6.5.m3.1.1.2.cmml" xref="S4.SS1.p6.5.m3.1.1.2">𝑖</ci><ci id="S4.SS1.p6.5.m3.1.1.3.cmml" xref="S4.SS1.p6.5.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.5.m3.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.5.m3.1d">italic_i < italic_j</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S4.F2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="223" id="S4.F2.g1" src="x2.png" width="409"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Discrete star graph.</figcaption> </figure> <div class="ltx_para" id="S4.SS1.p7"> <p class="ltx_p" id="S4.SS1.p7.1">As usual, the discrete Schrödinger operator <math alttext="H_{d}" class="ltx_Math" display="inline" id="S4.SS1.p7.1.m1.1"><semantics id="S4.SS1.p7.1.m1.1a"><msub id="S4.SS1.p7.1.m1.1.1" xref="S4.SS1.p7.1.m1.1.1.cmml"><mi id="S4.SS1.p7.1.m1.1.1.2" xref="S4.SS1.p7.1.m1.1.1.2.cmml">H</mi><mi id="S4.SS1.p7.1.m1.1.1.3" xref="S4.SS1.p7.1.m1.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.1.m1.1b"><apply id="S4.SS1.p7.1.m1.1.1.cmml" xref="S4.SS1.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p7.1.m1.1.1.1.cmml" xref="S4.SS1.p7.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p7.1.m1.1.1.2.cmml" xref="S4.SS1.p7.1.m1.1.1.2">𝐻</ci><ci id="S4.SS1.p7.1.m1.1.1.3.cmml" xref="S4.SS1.p7.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.1.m1.1c">H_{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.1.m1.1d">italic_H start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> on the graph corresponds to the negative second discrete derivative. It is defined component wise on each edge of the graph via</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx23"> <tbody id="S4.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})" class="ltx_Math" display="inline" id="S4.Ex15.m1.11"><semantics id="S4.Ex15.m1.11a"><mrow id="S4.Ex15.m1.11.11" xref="S4.Ex15.m1.11.11.cmml"><msubsup id="S4.Ex15.m1.11.11.3" xref="S4.Ex15.m1.11.11.3.cmml"><mi id="S4.Ex15.m1.11.11.3.2.2" mathvariant="normal" xref="S4.Ex15.m1.11.11.3.2.2.cmml">Ψ</mi><mrow id="S4.Ex15.m1.2.2.2.4" xref="S4.Ex15.m1.2.2.2.3.cmml"><mi id="S4.Ex15.m1.1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.cmml">i</mi><mo id="S4.Ex15.m1.2.2.2.4.1" xref="S4.Ex15.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex15.m1.2.2.2.2" xref="S4.Ex15.m1.2.2.2.2.cmml">j</mi></mrow><mrow id="S4.Ex15.m1.5.5.3.3" 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xref="S4.Ex15.m1.9.9.2.2.2.2">𝑗</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex15.m1.11c">\displaystyle\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.11d">roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mapsto" class="ltx_Math" display="inline" id="S4.Ex15.m2.1"><semantics id="S4.Ex15.m2.1a"><mo id="S4.Ex15.m2.1.1" stretchy="false" xref="S4.Ex15.m2.1.1.cmml">↦</mo><annotation-xml 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xref="S4.Ex16.m1.2.2.4.cmml">1</mn><msubsup id="S4.Ex16.m1.2.2.2" xref="S4.Ex16.m1.2.2.2.cmml"><mi id="S4.Ex16.m1.2.2.2.4.2" xref="S4.Ex16.m1.2.2.2.4.2.cmml">a</mi><mrow id="S4.Ex16.m1.2.2.2.2.2.4" xref="S4.Ex16.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex16.m1.1.1.1.1.1.1" xref="S4.Ex16.m1.1.1.1.1.1.1.cmml">i</mi><mo id="S4.Ex16.m1.2.2.2.2.2.4.1" xref="S4.Ex16.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.Ex16.m1.2.2.2.2.2.2" xref="S4.Ex16.m1.2.2.2.2.2.2.cmml">j</mi></mrow><mn id="S4.Ex16.m1.2.2.2.5" xref="S4.Ex16.m1.2.2.2.5.cmml">2</mn></msubsup></mfrac></mstyle></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex16.m1.2b"><apply id="S4.Ex16.m1.2.3.cmml" xref="S4.Ex16.m1.2.3"><minus id="S4.Ex16.m1.2.3.1.cmml" xref="S4.Ex16.m1.2.3"></minus><apply id="S4.Ex16.m1.2.2.cmml" xref="S4.Ex16.m1.2.2"><divide id="S4.Ex16.m1.2.2.3.cmml" xref="S4.Ex16.m1.2.2"></divide><cn id="S4.Ex16.m1.2.2.4.cmml" type="integer" xref="S4.Ex16.m1.2.2.4">1</cn><apply id="S4.Ex16.m1.2.2.2.cmml" xref="S4.Ex16.m1.2.2.2"><csymbol cd="ambiguous" id="S4.Ex16.m1.2.2.2.3.cmml" xref="S4.Ex16.m1.2.2.2">superscript</csymbol><apply id="S4.Ex16.m1.2.2.2.4.cmml" xref="S4.Ex16.m1.2.2.2"><csymbol cd="ambiguous" id="S4.Ex16.m1.2.2.2.4.1.cmml" xref="S4.Ex16.m1.2.2.2">subscript</csymbol><ci id="S4.Ex16.m1.2.2.2.4.2.cmml" xref="S4.Ex16.m1.2.2.2.4.2">𝑎</ci><list id="S4.Ex16.m1.2.2.2.2.2.3.cmml" xref="S4.Ex16.m1.2.2.2.2.2.4"><ci id="S4.Ex16.m1.1.1.1.1.1.1.cmml" xref="S4.Ex16.m1.1.1.1.1.1.1">𝑖</ci><ci id="S4.Ex16.m1.2.2.2.2.2.2.cmml" xref="S4.Ex16.m1.2.2.2.2.2.2">𝑗</ci></list></apply><cn id="S4.Ex16.m1.2.2.2.5.cmml" type="integer" xref="S4.Ex16.m1.2.2.2.5">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex16.m1.2c">\displaystyle-\frac{1}{a_{i,j}^{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m1.2d">- divide start_ARG 1 end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\bigg{[}\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})}-a_{i,j})" class="ltx_math_unparsed" display="inline" id="S4.Ex16.m2.12"><semantics id="S4.Ex16.m2.12a"><mrow id="S4.Ex16.m2.12b"><mo id="S4.Ex16.m2.12.13" maxsize="210%" minsize="210%">[</mo><msubsup id="S4.Ex16.m2.12.14"><mi id="S4.Ex16.m2.12.14.2.2" mathvariant="normal">Ψ</mi><mrow id="S4.Ex16.m2.2.2.2.4"><mi id="S4.Ex16.m2.1.1.1.1">i</mi><mo id="S4.Ex16.m2.2.2.2.4.1">,</mo><mi id="S4.Ex16.m2.2.2.2.2">j</mi></mrow><mrow id="S4.Ex16.m2.5.5.3.3"><mo id="S4.Ex16.m2.5.5.3.3.2" stretchy="false">(</mo><msub id="S4.Ex16.m2.5.5.3.3.1"><mi id="S4.Ex16.m2.5.5.3.3.1.2">a</mi><mrow id="S4.Ex16.m2.4.4.2.2.2.4"><mi id="S4.Ex16.m2.3.3.1.1.1.1">i</mi><mo id="S4.Ex16.m2.4.4.2.2.2.4.1">,</mo><mi id="S4.Ex16.m2.4.4.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex16.m2.5.5.3.3.3" stretchy="false">)</mo></mrow></msubsup><mrow id="S4.Ex16.m2.12.15"><mo id="S4.Ex16.m2.12.15.1" stretchy="false">(</mo><msubsup id="S4.Ex16.m2.12.15.2"><mi id="S4.Ex16.m2.12.15.2.2.2">x</mi><msub id="S4.Ex16.m2.7.7.2"><mi id="S4.Ex16.m2.7.7.2.4">n</mi><mrow id="S4.Ex16.m2.7.7.2.2.2.4"><mi id="S4.Ex16.m2.6.6.1.1.1.1">i</mi><mo id="S4.Ex16.m2.7.7.2.2.2.4.1">,</mo><mi id="S4.Ex16.m2.7.7.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex16.m2.10.10.3.3"><mo id="S4.Ex16.m2.10.10.3.3.2" stretchy="false">(</mo><msub id="S4.Ex16.m2.10.10.3.3.1"><mi id="S4.Ex16.m2.10.10.3.3.1.2">a</mi><mrow id="S4.Ex16.m2.9.9.2.2.2.4"><mi id="S4.Ex16.m2.8.8.1.1.1.1">i</mi><mo id="S4.Ex16.m2.9.9.2.2.2.4.1">,</mo><mi id="S4.Ex16.m2.9.9.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex16.m2.10.10.3.3.3" stretchy="false">)</mo></mrow></msubsup><mo id="S4.Ex16.m2.12.15.3">−</mo><msub id="S4.Ex16.m2.12.15.4"><mi id="S4.Ex16.m2.12.15.4.2">a</mi><mrow id="S4.Ex16.m2.12.12.2.4"><mi id="S4.Ex16.m2.11.11.1.1">i</mi><mo id="S4.Ex16.m2.12.12.2.4.1">,</mo><mi id="S4.Ex16.m2.12.12.2.2">j</mi></mrow></msub><mo id="S4.Ex16.m2.12.15.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex16.m2.12c">\displaystyle\bigg{[}\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})}-a_{i,j})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m2.12d">[ roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT - italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-2\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})+\Psi_{i,j}^{(a_% {i,j})}(x_{n_{i,j}}^{(a_{i,j})}+a_{i,j})\bigg{]}." class="ltx_math_unparsed" display="inline" id="S4.Ex17.m1.22"><semantics id="S4.Ex17.m1.22a"><mrow id="S4.Ex17.m1.22b"><mo id="S4.Ex17.m1.22.23">−</mo><mn id="S4.Ex17.m1.22.24">2</mn><msubsup id="S4.Ex17.m1.22.25"><mi id="S4.Ex17.m1.22.25.2.2" mathvariant="normal">Ψ</mi><mrow id="S4.Ex17.m1.2.2.2.4"><mi id="S4.Ex17.m1.1.1.1.1">i</mi><mo id="S4.Ex17.m1.2.2.2.4.1">,</mo><mi id="S4.Ex17.m1.2.2.2.2">j</mi></mrow><mrow id="S4.Ex17.m1.5.5.3.3"><mo id="S4.Ex17.m1.5.5.3.3.2" stretchy="false">(</mo><msub id="S4.Ex17.m1.5.5.3.3.1"><mi id="S4.Ex17.m1.5.5.3.3.1.2">a</mi><mrow id="S4.Ex17.m1.4.4.2.2.2.4"><mi id="S4.Ex17.m1.3.3.1.1.1.1">i</mi><mo id="S4.Ex17.m1.4.4.2.2.2.4.1">,</mo><mi id="S4.Ex17.m1.4.4.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex17.m1.5.5.3.3.3" stretchy="false">)</mo></mrow></msubsup><mrow id="S4.Ex17.m1.22.26"><mo id="S4.Ex17.m1.22.26.1" stretchy="false">(</mo><msubsup id="S4.Ex17.m1.22.26.2"><mi id="S4.Ex17.m1.22.26.2.2.2">x</mi><msub id="S4.Ex17.m1.7.7.2"><mi id="S4.Ex17.m1.7.7.2.4">n</mi><mrow id="S4.Ex17.m1.7.7.2.2.2.4"><mi id="S4.Ex17.m1.6.6.1.1.1.1">i</mi><mo id="S4.Ex17.m1.7.7.2.2.2.4.1">,</mo><mi id="S4.Ex17.m1.7.7.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex17.m1.10.10.3.3"><mo id="S4.Ex17.m1.10.10.3.3.2" stretchy="false">(</mo><msub id="S4.Ex17.m1.10.10.3.3.1"><mi id="S4.Ex17.m1.10.10.3.3.1.2">a</mi><mrow id="S4.Ex17.m1.9.9.2.2.2.4"><mi id="S4.Ex17.m1.8.8.1.1.1.1">i</mi><mo id="S4.Ex17.m1.9.9.2.2.2.4.1">,</mo><mi id="S4.Ex17.m1.9.9.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex17.m1.10.10.3.3.3" stretchy="false">)</mo></mrow></msubsup><mo id="S4.Ex17.m1.22.26.3" stretchy="false">)</mo></mrow><mo id="S4.Ex17.m1.22.27">+</mo><msubsup id="S4.Ex17.m1.22.28"><mi id="S4.Ex17.m1.22.28.2.2" mathvariant="normal">Ψ</mi><mrow id="S4.Ex17.m1.12.12.2.4"><mi id="S4.Ex17.m1.11.11.1.1">i</mi><mo id="S4.Ex17.m1.12.12.2.4.1">,</mo><mi id="S4.Ex17.m1.12.12.2.2">j</mi></mrow><mrow id="S4.Ex17.m1.15.15.3.3"><mo id="S4.Ex17.m1.15.15.3.3.2" stretchy="false">(</mo><msub id="S4.Ex17.m1.15.15.3.3.1"><mi id="S4.Ex17.m1.15.15.3.3.1.2">a</mi><mrow id="S4.Ex17.m1.14.14.2.2.2.4"><mi id="S4.Ex17.m1.13.13.1.1.1.1">i</mi><mo id="S4.Ex17.m1.14.14.2.2.2.4.1">,</mo><mi id="S4.Ex17.m1.14.14.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex17.m1.15.15.3.3.3" stretchy="false">)</mo></mrow></msubsup><mrow id="S4.Ex17.m1.22.29"><mo id="S4.Ex17.m1.22.29.1" stretchy="false">(</mo><msubsup id="S4.Ex17.m1.22.29.2"><mi id="S4.Ex17.m1.22.29.2.2.2">x</mi><msub id="S4.Ex17.m1.17.17.2"><mi id="S4.Ex17.m1.17.17.2.4">n</mi><mrow id="S4.Ex17.m1.17.17.2.2.2.4"><mi id="S4.Ex17.m1.16.16.1.1.1.1">i</mi><mo id="S4.Ex17.m1.17.17.2.2.2.4.1">,</mo><mi id="S4.Ex17.m1.17.17.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex17.m1.20.20.3.3"><mo id="S4.Ex17.m1.20.20.3.3.2" stretchy="false">(</mo><msub id="S4.Ex17.m1.20.20.3.3.1"><mi id="S4.Ex17.m1.20.20.3.3.1.2">a</mi><mrow id="S4.Ex17.m1.19.19.2.2.2.4"><mi id="S4.Ex17.m1.18.18.1.1.1.1">i</mi><mo id="S4.Ex17.m1.19.19.2.2.2.4.1">,</mo><mi id="S4.Ex17.m1.19.19.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex17.m1.20.20.3.3.3" stretchy="false">)</mo></mrow></msubsup><mo id="S4.Ex17.m1.22.29.3">+</mo><msub id="S4.Ex17.m1.22.29.4"><mi id="S4.Ex17.m1.22.29.4.2">a</mi><mrow id="S4.Ex17.m1.22.22.2.4"><mi id="S4.Ex17.m1.21.21.1.1">i</mi><mo id="S4.Ex17.m1.22.22.2.4.1">,</mo><mi id="S4.Ex17.m1.22.22.2.2">j</mi></mrow></msub><mo id="S4.Ex17.m1.22.29.5" stretchy="false">)</mo></mrow><mo id="S4.Ex17.m1.22.30" maxsize="210%" minsize="210%">]</mo><mo id="S4.Ex17.m1.22.31" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex17.m1.22c">\displaystyle-2\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})+\Psi_{i,j}^{(a_% {i,j})}(x_{n_{i,j}}^{(a_{i,j})}+a_{i,j})\bigg{]}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex17.m1.22d">- 2 roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) + roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p8"> <p class="ltx_p" id="S4.SS1.p8.1">It satisfies</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx24"> <tbody id="S4.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\langle H_{d}\Psi,\Phi\rangle-\langle" class="ltx_math_unparsed" display="inline" id="S4.Ex18.m1.1"><semantics id="S4.Ex18.m1.1a"><mrow id="S4.Ex18.m1.1b"><mrow id="S4.Ex18.m1.1.2"><mo id="S4.Ex18.m1.1.2.1" stretchy="false">⟨</mo><msub id="S4.Ex18.m1.1.2.2"><mi id="S4.Ex18.m1.1.2.2.2">H</mi><mi id="S4.Ex18.m1.1.2.2.3">d</mi></msub><mi id="S4.Ex18.m1.1.2.3" mathvariant="normal">Ψ</mi><mo id="S4.Ex18.m1.1.2.4">,</mo><mi id="S4.Ex18.m1.1.1" mathvariant="normal">Φ</mi><mo id="S4.Ex18.m1.1.2.5" stretchy="false">⟩</mo></mrow><mo id="S4.Ex18.m1.1.3">−</mo><mo id="S4.Ex18.m1.1.4" stretchy="false">⟨</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex18.m1.1c">\displaystyle\langle H_{d}\Psi,\Phi\rangle-\langle</annotation><annotation encoding="application/x-llamapun" id="S4.Ex18.m1.1d">⟨ italic_H start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT roman_Ψ , roman_Φ ⟩ - ⟨</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\Psi,H_{d}\Phi\rangle=\sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{V}C_{i,j}\bigg{[}\Psi_{i,j}^{(a_{i,j})}(0)\Phi_{i,j}^{(a_{% i,j})*}(a_{i,j})" class="ltx_math_unparsed" display="inline" id="S4.Ex18.m2.17"><semantics id="S4.Ex18.m2.17a"><mrow id="S4.Ex18.m2.17b"><mi id="S4.Ex18.m2.17.17" mathvariant="normal">Ψ</mi><mo id="S4.Ex18.m2.17.18">,</mo><msub id="S4.Ex18.m2.17.19"><mi id="S4.Ex18.m2.17.19.2">H</mi><mi id="S4.Ex18.m2.17.19.3">d</mi></msub><mi id="S4.Ex18.m2.17.20" mathvariant="normal">Φ</mi><mo id="S4.Ex18.m2.17.21" stretchy="false">⟩</mo><mo id="S4.Ex18.m2.17.22">=</mo><mstyle displaystyle="true" id="S4.Ex18.m2.17.23"><mo id="S4.Ex18.m2.17.23a" movablelimits="false">∑</mo></mstyle><msub id="S4.Ex18.m2.1.1"><mi id="S4.Ex18.m2.1.1a"></mi><mtable id="S4.Ex18.m2.1.1.1.1.1.1" rowspacing="0pt"><mtr id="S4.Ex18.m2.1.1.1.1.1.1a"><mtd id="S4.Ex18.m2.1.1.1.1.1.1b"><mrow id="S4.Ex18.m2.1.1.1.1.1.1.2.2.2.2"><mrow id="S4.Ex18.m2.1.1.1.1.1.1.2.2.2.2.4.2"><mi id="S4.Ex18.m2.1.1.1.1.1.1.1.1.1.1.1">i</mi><mo id="S4.Ex18.m2.1.1.1.1.1.1.2.2.2.2.4.2.1">,</mo><mi id="S4.Ex18.m2.1.1.1.1.1.1.2.2.2.2.2">j</mi></mrow><mo id="S4.Ex18.m2.1.1.1.1.1.1.2.2.2.2.3">=</mo><mn id="S4.Ex18.m2.1.1.1.1.1.1.2.2.2.2.5">1</mn></mrow></mtd></mtr><mtr id="S4.Ex18.m2.1.1.1.1.1.1c"><mtd id="S4.Ex18.m2.1.1.1.1.1.1d"><mrow id="S4.Ex18.m2.1.1.1.1.1.1.3.1.1"><mi id="S4.Ex18.m2.1.1.1.1.1.1.3.1.1.2">i</mi><mo id="S4.Ex18.m2.1.1.1.1.1.1.3.1.1.1"><</mo><mi id="S4.Ex18.m2.1.1.1.1.1.1.3.1.1.3">j</mi></mrow></mtd></mtr></mtable></msub><msup id="S4.Ex18.m2.17.24"><mi id="S4.Ex18.m2.17.24a"></mi><mi id="S4.Ex18.m2.17.24.1">V</mi></msup><mi id="S4.Ex18.m2.17.25">C</mi><msub id="S4.Ex18.m2.3.3"><mi id="S4.Ex18.m2.3.3a"></mi><mrow id="S4.Ex18.m2.3.3.2.4"><mi id="S4.Ex18.m2.2.2.1.1">i</mi><mo id="S4.Ex18.m2.3.3.2.4.1">,</mo><mi id="S4.Ex18.m2.3.3.2.2">j</mi></mrow></msub><mo id="S4.Ex18.m2.17.26" maxsize="210%" minsize="210%">[</mo><msubsup id="S4.Ex18.m2.17.27"><mi id="S4.Ex18.m2.17.27.2.2" mathvariant="normal">Ψ</mi><mrow id="S4.Ex18.m2.5.5.2.4"><mi id="S4.Ex18.m2.4.4.1.1">i</mi><mo id="S4.Ex18.m2.5.5.2.4.1">,</mo><mi id="S4.Ex18.m2.5.5.2.2">j</mi></mrow><mrow id="S4.Ex18.m2.8.8.3.3"><mo id="S4.Ex18.m2.8.8.3.3.2" stretchy="false">(</mo><msub id="S4.Ex18.m2.8.8.3.3.1"><mi id="S4.Ex18.m2.8.8.3.3.1.2">a</mi><mrow id="S4.Ex18.m2.7.7.2.2.2.4"><mi id="S4.Ex18.m2.6.6.1.1.1.1">i</mi><mo id="S4.Ex18.m2.7.7.2.2.2.4.1">,</mo><mi id="S4.Ex18.m2.7.7.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex18.m2.8.8.3.3.3" stretchy="false">)</mo></mrow></msubsup><mrow id="S4.Ex18.m2.17.28"><mo id="S4.Ex18.m2.17.28.1" stretchy="false">(</mo><mn id="S4.Ex18.m2.17.28.2">0</mn><mo id="S4.Ex18.m2.17.28.3" stretchy="false">)</mo></mrow><msubsup id="S4.Ex18.m2.17.29"><mi id="S4.Ex18.m2.17.29.2.2" mathvariant="normal">Φ</mi><mrow id="S4.Ex18.m2.10.10.2.4"><mi id="S4.Ex18.m2.9.9.1.1">i</mi><mo id="S4.Ex18.m2.10.10.2.4.1">,</mo><mi id="S4.Ex18.m2.10.10.2.2">j</mi></mrow><mrow id="S4.Ex18.m2.14.14.4.4"><mrow id="S4.Ex18.m2.14.14.4.4.1.1"><mo id="S4.Ex18.m2.14.14.4.4.1.1.2" stretchy="false">(</mo><msub id="S4.Ex18.m2.14.14.4.4.1.1.1"><mi id="S4.Ex18.m2.14.14.4.4.1.1.1.2">a</mi><mrow id="S4.Ex18.m2.12.12.2.2.2.4"><mi id="S4.Ex18.m2.11.11.1.1.1.1">i</mi><mo id="S4.Ex18.m2.12.12.2.2.2.4.1">,</mo><mi id="S4.Ex18.m2.12.12.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex18.m2.14.14.4.4.1.1.3" stretchy="false">)</mo></mrow><mo id="S4.Ex18.m2.14.14.4.4.2" lspace="0.222em">⁣</mo><mo id="S4.Ex18.m2.13.13.3.3">∗</mo></mrow></msubsup><mrow id="S4.Ex18.m2.17.30"><mo id="S4.Ex18.m2.17.30.1" stretchy="false">(</mo><msub id="S4.Ex18.m2.17.30.2"><mi id="S4.Ex18.m2.17.30.2.2">a</mi><mrow id="S4.Ex18.m2.16.16.2.4"><mi id="S4.Ex18.m2.15.15.1.1">i</mi><mo id="S4.Ex18.m2.16.16.2.4.1">,</mo><mi id="S4.Ex18.m2.16.16.2.2">j</mi></mrow></msub><mo id="S4.Ex18.m2.17.30.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex18.m2.17c">\displaystyle\Psi,H_{d}\Phi\rangle=\sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{V}C_{i,j}\bigg{[}\Psi_{i,j}^{(a_{i,j})}(0)\Phi_{i,j}^{(a_{% i,j})*}(a_{i,j})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex18.m2.17d">roman_Ψ , italic_H start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT roman_Φ ⟩ = ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_i , italic_j = 1 end_CELL end_ROW start_ROW start_CELL italic_i < italic_j end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [ roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( 0 ) roman_Φ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ∗ end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S4.Ex19.m1.13.13.4.4.1.1.1.2.cmml" xref="S4.Ex19.m1.13.13.4.4.1.1.1.2">𝑎</ci><list id="S4.Ex19.m1.11.11.2.2.2.3.cmml" xref="S4.Ex19.m1.11.11.2.2.2.4"><ci id="S4.Ex19.m1.10.10.1.1.1.1.cmml" xref="S4.Ex19.m1.10.10.1.1.1.1">𝑖</ci><ci id="S4.Ex19.m1.11.11.2.2.2.2.cmml" xref="S4.Ex19.m1.11.11.2.2.2.2">𝑗</ci></list></apply><times id="S4.Ex19.m1.12.12.3.3.cmml" xref="S4.Ex19.m1.12.12.3.3"></times></list></apply><cn id="S4.Ex19.m1.14.14.cmml" type="integer" xref="S4.Ex19.m1.14.14">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex19.m1.15c">\displaystyle-\Psi_{i,j}^{(a_{i,j})}(a_{i,j})\Phi_{i,j}^{(a_{i,j})*}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex19.m1.15d">- roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) roman_Φ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ∗ end_POSTSUPERSCRIPT ( 0 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\Psi_{i,j}^{(a_{i,j})}(L_{i,j})\Phi_{i,j}^{(a_{i,j})*}(L_{i,j}-a% _{i,j})" class="ltx_Math" display="inline" id="S4.Ex20.m1.19"><semantics id="S4.Ex20.m1.19a"><mrow id="S4.Ex20.m1.19.19" xref="S4.Ex20.m1.19.19.cmml"><mo id="S4.Ex20.m1.19.19a" xref="S4.Ex20.m1.19.19.cmml">+</mo><mrow id="S4.Ex20.m1.19.19.2" xref="S4.Ex20.m1.19.19.2.cmml"><msubsup id="S4.Ex20.m1.19.19.2.4" xref="S4.Ex20.m1.19.19.2.4.cmml"><mi id="S4.Ex20.m1.19.19.2.4.2.2" mathvariant="normal" 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start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\Psi_{i,j}^{(a_{i,j})}(L_{i,j}-a_{i,j})\Phi_{i,j}^{(a_{i,j})*}(L% _{i,j})\bigg{]}." class="ltx_math_unparsed" display="inline" id="S4.E30.m2.17"><semantics id="S4.E30.m2.17a"><mrow id="S4.E30.m2.17b"><mo id="S4.E30.m2.17.18">−</mo><msubsup id="S4.E30.m2.17.19"><mi id="S4.E30.m2.17.19.2.2" mathvariant="normal">Ψ</mi><mrow id="S4.E30.m2.2.2.2.4"><mi id="S4.E30.m2.1.1.1.1">i</mi><mo id="S4.E30.m2.2.2.2.4.1">,</mo><mi id="S4.E30.m2.2.2.2.2">j</mi></mrow><mrow id="S4.E30.m2.5.5.3.3"><mo id="S4.E30.m2.5.5.3.3.2" stretchy="false">(</mo><msub id="S4.E30.m2.5.5.3.3.1"><mi 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id="S4.E30.m2.11.11.2.4.1">,</mo><mi id="S4.E30.m2.11.11.2.2">j</mi></mrow><mrow id="S4.E30.m2.15.15.4.4"><mrow id="S4.E30.m2.15.15.4.4.1.1"><mo id="S4.E30.m2.15.15.4.4.1.1.2" stretchy="false">(</mo><msub id="S4.E30.m2.15.15.4.4.1.1.1"><mi id="S4.E30.m2.15.15.4.4.1.1.1.2">a</mi><mrow id="S4.E30.m2.13.13.2.2.2.4"><mi id="S4.E30.m2.12.12.1.1.1.1">i</mi><mo id="S4.E30.m2.13.13.2.2.2.4.1">,</mo><mi id="S4.E30.m2.13.13.2.2.2.2">j</mi></mrow></msub><mo id="S4.E30.m2.15.15.4.4.1.1.3" stretchy="false">)</mo></mrow><mo id="S4.E30.m2.15.15.4.4.2" lspace="0.222em">⁣</mo><mo id="S4.E30.m2.14.14.3.3">∗</mo></mrow></msubsup><mrow id="S4.E30.m2.17.22"><mo id="S4.E30.m2.17.22.1" stretchy="false">(</mo><msub id="S4.E30.m2.17.22.2"><mi id="S4.E30.m2.17.22.2.2">L</mi><mrow id="S4.E30.m2.17.17.2.4"><mi id="S4.E30.m2.16.16.1.1">i</mi><mo id="S4.E30.m2.17.17.2.4.1">,</mo><mi id="S4.E30.m2.17.17.2.2">j</mi></mrow></msub><mo id="S4.E30.m2.17.22.3" stretchy="false">)</mo></mrow><mo id="S4.E30.m2.17.23" maxsize="210%" minsize="210%">]</mo><mo id="S4.E30.m2.17.24" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S4.E30.m2.17c">\displaystyle-\Psi_{i,j}^{(a_{i,j})}(L_{i,j}-a_{i,j})\Phi_{i,j}^{(a_{i,j})*}(L% _{i,j})\bigg{]}.</annotation><annotation encoding="application/x-llamapun" id="S4.E30.m2.17d">- roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) roman_Φ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ∗ end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(30)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p9"> <p class="ltx_p" id="S4.SS1.p9.1">In fact, <math alttext="H_{d}" class="ltx_Math" display="inline" id="S4.SS1.p9.1.m1.1"><semantics id="S4.SS1.p9.1.m1.1a"><msub id="S4.SS1.p9.1.m1.1.1" xref="S4.SS1.p9.1.m1.1.1.cmml"><mi id="S4.SS1.p9.1.m1.1.1.2" xref="S4.SS1.p9.1.m1.1.1.2.cmml">H</mi><mi id="S4.SS1.p9.1.m1.1.1.3" xref="S4.SS1.p9.1.m1.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p9.1.m1.1b"><apply id="S4.SS1.p9.1.m1.1.1.cmml" xref="S4.SS1.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p9.1.m1.1.1.1.cmml" xref="S4.SS1.p9.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p9.1.m1.1.1.2.cmml" xref="S4.SS1.p9.1.m1.1.1.2">𝐻</ci><ci id="S4.SS1.p9.1.m1.1.1.3.cmml" xref="S4.SS1.p9.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p9.1.m1.1c">H_{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p9.1.m1.1d">italic_H start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> is a matrix, and, hence, it is self-adjoint if and only if (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.E30" title="In 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">30</span></a>) is zero. Similar as in the previous section, we impose a continuity condition and some current conservation rule. The continuity condition reads as</p> <table class="ltx_equation ltx_eqn_table" id="S4.E31"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Psi_{i,j}^{(a_{i,j})}(0)=\phi_{i},\quad\Psi_{i,j}^{(a_{i,j})}(L_{i,j})=\phi_{% j}," class="ltx_Math" display="block" id="S4.E31.m1.14"><semantics id="S4.E31.m1.14a"><mrow id="S4.E31.m1.14.14.1"><mrow id="S4.E31.m1.14.14.1.1.2" xref="S4.E31.m1.14.14.1.1.3.cmml"><mrow id="S4.E31.m1.14.14.1.1.1.1" xref="S4.E31.m1.14.14.1.1.1.1.cmml"><mrow id="S4.E31.m1.14.14.1.1.1.1.2" xref="S4.E31.m1.14.14.1.1.1.1.2.cmml"><msubsup id="S4.E31.m1.14.14.1.1.1.1.2.2" xref="S4.E31.m1.14.14.1.1.1.1.2.2.cmml"><mi id="S4.E31.m1.14.14.1.1.1.1.2.2.2.2" mathvariant="normal" xref="S4.E31.m1.14.14.1.1.1.1.2.2.2.2.cmml">Ψ</mi><mrow id="S4.E31.m1.2.2.2.4" xref="S4.E31.m1.2.2.2.3.cmml"><mi id="S4.E31.m1.1.1.1.1" xref="S4.E31.m1.1.1.1.1.cmml">i</mi><mo 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xref="S4.E31.m1.14.14.1.1.2.2.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E31.m1.14c">\Psi_{i,j}^{(a_{i,j})}(0)=\phi_{i},\quad\Psi_{i,j}^{(a_{i,j})}(L_{i,j})=\phi_{% j},</annotation><annotation encoding="application/x-llamapun" id="S4.E31.m1.14d">roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( 0 ) = italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(31)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p9.4">for some complex numbers <math alttext="\phi_{i}" class="ltx_Math" display="inline" id="S4.SS1.p9.2.m1.1"><semantics id="S4.SS1.p9.2.m1.1a"><msub id="S4.SS1.p9.2.m1.1.1" xref="S4.SS1.p9.2.m1.1.1.cmml"><mi id="S4.SS1.p9.2.m1.1.1.2" xref="S4.SS1.p9.2.m1.1.1.2.cmml">ϕ</mi><mi id="S4.SS1.p9.2.m1.1.1.3" xref="S4.SS1.p9.2.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p9.2.m1.1b"><apply id="S4.SS1.p9.2.m1.1.1.cmml" xref="S4.SS1.p9.2.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p9.2.m1.1.1.1.cmml" xref="S4.SS1.p9.2.m1.1.1">subscript</csymbol><ci id="S4.SS1.p9.2.m1.1.1.2.cmml" xref="S4.SS1.p9.2.m1.1.1.2">italic-ϕ</ci><ci id="S4.SS1.p9.2.m1.1.1.3.cmml" xref="S4.SS1.p9.2.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p9.2.m1.1c">\phi_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p9.2.m1.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="i<j" class="ltx_Math" display="inline" id="S4.SS1.p9.3.m2.1"><semantics id="S4.SS1.p9.3.m2.1a"><mrow id="S4.SS1.p9.3.m2.1.1" xref="S4.SS1.p9.3.m2.1.1.cmml"><mi id="S4.SS1.p9.3.m2.1.1.2" xref="S4.SS1.p9.3.m2.1.1.2.cmml">i</mi><mo id="S4.SS1.p9.3.m2.1.1.1" xref="S4.SS1.p9.3.m2.1.1.1.cmml"><</mo><mi id="S4.SS1.p9.3.m2.1.1.3" xref="S4.SS1.p9.3.m2.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p9.3.m2.1b"><apply id="S4.SS1.p9.3.m2.1.1.cmml" xref="S4.SS1.p9.3.m2.1.1"><lt id="S4.SS1.p9.3.m2.1.1.1.cmml" xref="S4.SS1.p9.3.m2.1.1.1"></lt><ci id="S4.SS1.p9.3.m2.1.1.2.cmml" xref="S4.SS1.p9.3.m2.1.1.2">𝑖</ci><ci id="S4.SS1.p9.3.m2.1.1.3.cmml" xref="S4.SS1.p9.3.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p9.3.m2.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p9.3.m2.1d">italic_i < italic_j</annotation></semantics></math> and <math alttext="C_{i,j}=1" class="ltx_Math" display="inline" id="S4.SS1.p9.4.m3.2"><semantics id="S4.SS1.p9.4.m3.2a"><mrow id="S4.SS1.p9.4.m3.2.3" xref="S4.SS1.p9.4.m3.2.3.cmml"><msub id="S4.SS1.p9.4.m3.2.3.2" xref="S4.SS1.p9.4.m3.2.3.2.cmml"><mi id="S4.SS1.p9.4.m3.2.3.2.2" xref="S4.SS1.p9.4.m3.2.3.2.2.cmml">C</mi><mrow id="S4.SS1.p9.4.m3.2.2.2.4" xref="S4.SS1.p9.4.m3.2.2.2.3.cmml"><mi id="S4.SS1.p9.4.m3.1.1.1.1" xref="S4.SS1.p9.4.m3.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p9.4.m3.2.2.2.4.1" xref="S4.SS1.p9.4.m3.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p9.4.m3.2.2.2.2" xref="S4.SS1.p9.4.m3.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p9.4.m3.2.3.1" xref="S4.SS1.p9.4.m3.2.3.1.cmml">=</mo><mn id="S4.SS1.p9.4.m3.2.3.3" xref="S4.SS1.p9.4.m3.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p9.4.m3.2b"><apply id="S4.SS1.p9.4.m3.2.3.cmml" xref="S4.SS1.p9.4.m3.2.3"><eq id="S4.SS1.p9.4.m3.2.3.1.cmml" xref="S4.SS1.p9.4.m3.2.3.1"></eq><apply id="S4.SS1.p9.4.m3.2.3.2.cmml" xref="S4.SS1.p9.4.m3.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p9.4.m3.2.3.2.1.cmml" xref="S4.SS1.p9.4.m3.2.3.2">subscript</csymbol><ci id="S4.SS1.p9.4.m3.2.3.2.2.cmml" xref="S4.SS1.p9.4.m3.2.3.2.2">𝐶</ci><list id="S4.SS1.p9.4.m3.2.2.2.3.cmml" xref="S4.SS1.p9.4.m3.2.2.2.4"><ci id="S4.SS1.p9.4.m3.1.1.1.1.cmml" xref="S4.SS1.p9.4.m3.1.1.1.1">𝑖</ci><ci id="S4.SS1.p9.4.m3.2.2.2.2.cmml" xref="S4.SS1.p9.4.m3.2.2.2.2">𝑗</ci></list></apply><cn id="S4.SS1.p9.4.m3.2.3.3.cmml" type="integer" xref="S4.SS1.p9.4.m3.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p9.4.m3.2c">C_{i,j}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p9.4.m3.2d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>. The discrete version of the current conservation</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx25"> <tbody id="S4.E32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}-&\sum_{j<i}C_{i,j}\left[\Psi_{j,i}^{(a_{j,i})}(L_{j% ,i})-\Psi_{j,i}^{(a_{j,i})}(L_{j,i}-a_{j,i})\right]\\ &+\sum_{j>i}C_{i,j}\left[\Psi_{i,j}^{(a_{i,j})}(a_{i,j})-\Psi_{i,j}^{(a_{i,j})% }(0)\right]=\lambda_{i}\phi_{i},\end{split}" class="ltx_Math" display="inline" id="S4.E32.m1.55"><semantics id="S4.E32.m1.55a"><mtable columnspacing="0pt" id="S4.E32.m1.55.55.3" rowspacing="0pt"><mtr id="S4.E32.m1.55.55.3a"><mtd class="ltx_align_right" columnalign="right" id="S4.E32.m1.55.55.3b"><mo id="S4.E32.m1.1.1.1.1.1.1" xref="S4.E32.m1.1.1.1.1.1.1.cmml">−</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E32.m1.55.55.3c"><mrow 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id="S4.E32.m1.55c">\displaystyle\begin{split}-&\sum_{j<i}C_{i,j}\left[\Psi_{j,i}^{(a_{j,i})}(L_{j% ,i})-\Psi_{j,i}^{(a_{j,i})}(L_{j,i}-a_{j,i})\right]\\ &+\sum_{j>i}C_{i,j}\left[\Psi_{i,j}^{(a_{i,j})}(a_{i,j})-\Psi_{i,j}^{(a_{i,j})% }(0)\right]=\lambda_{i}\phi_{i},\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E32.m1.55d">start_ROW start_CELL - end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_j < italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [ roman_Ψ start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT ) - roman_Ψ start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_j , italic_i end_POSTSUBSCRIPT ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_j > italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [ roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) - roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( 0 ) ] = italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(32)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p9.7">for some real parameters <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S4.SS1.p9.5.m1.1"><semantics id="S4.SS1.p9.5.m1.1a"><msub id="S4.SS1.p9.5.m1.1.1" xref="S4.SS1.p9.5.m1.1.1.cmml"><mi id="S4.SS1.p9.5.m1.1.1.2" xref="S4.SS1.p9.5.m1.1.1.2.cmml">λ</mi><mi id="S4.SS1.p9.5.m1.1.1.3" xref="S4.SS1.p9.5.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p9.5.m1.1b"><apply id="S4.SS1.p9.5.m1.1.1.cmml" xref="S4.SS1.p9.5.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p9.5.m1.1.1.1.cmml" xref="S4.SS1.p9.5.m1.1.1">subscript</csymbol><ci id="S4.SS1.p9.5.m1.1.1.2.cmml" xref="S4.SS1.p9.5.m1.1.1.2">𝜆</ci><ci id="S4.SS1.p9.5.m1.1.1.3.cmml" xref="S4.SS1.p9.5.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p9.5.m1.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p9.5.m1.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p9.6.m2.1"><semantics id="S4.SS1.p9.6.m2.1a"><mi id="S4.SS1.p9.6.m2.1.1" xref="S4.SS1.p9.6.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p9.6.m2.1b"><ci id="S4.SS1.p9.6.m2.1.1.cmml" xref="S4.SS1.p9.6.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p9.6.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p9.6.m2.1d">italic_i</annotation></semantics></math> in <math alttext="\{1,...,V\}" class="ltx_Math" display="inline" id="S4.SS1.p9.7.m3.3"><semantics id="S4.SS1.p9.7.m3.3a"><mrow id="S4.SS1.p9.7.m3.3.4.2" xref="S4.SS1.p9.7.m3.3.4.1.cmml"><mo id="S4.SS1.p9.7.m3.3.4.2.1" stretchy="false" xref="S4.SS1.p9.7.m3.3.4.1.cmml">{</mo><mn id="S4.SS1.p9.7.m3.1.1" xref="S4.SS1.p9.7.m3.1.1.cmml">1</mn><mo id="S4.SS1.p9.7.m3.3.4.2.2" xref="S4.SS1.p9.7.m3.3.4.1.cmml">,</mo><mi id="S4.SS1.p9.7.m3.2.2" mathvariant="normal" xref="S4.SS1.p9.7.m3.2.2.cmml">…</mi><mo id="S4.SS1.p9.7.m3.3.4.2.3" xref="S4.SS1.p9.7.m3.3.4.1.cmml">,</mo><mi id="S4.SS1.p9.7.m3.3.3" xref="S4.SS1.p9.7.m3.3.3.cmml">V</mi><mo id="S4.SS1.p9.7.m3.3.4.2.4" stretchy="false" xref="S4.SS1.p9.7.m3.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p9.7.m3.3b"><set id="S4.SS1.p9.7.m3.3.4.1.cmml" xref="S4.SS1.p9.7.m3.3.4.2"><cn id="S4.SS1.p9.7.m3.1.1.cmml" type="integer" xref="S4.SS1.p9.7.m3.1.1">1</cn><ci id="S4.SS1.p9.7.m3.2.2.cmml" xref="S4.SS1.p9.7.m3.2.2">…</ci><ci id="S4.SS1.p9.7.m3.3.3.cmml" xref="S4.SS1.p9.7.m3.3.3">𝑉</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p9.7.m3.3c">\{1,...,V\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p9.7.m3.3d">{ 1 , … , italic_V }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p10"> <p class="ltx_p" id="S4.SS1.p10.2">Inspired from Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E3" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">3</span></a>) we use a somehow different linear combination of the fundamental solution in (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E3" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">3</span></a>) which is adapted to the above continuity and current conservation law similar in (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E24" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">24</span></a>) as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx26"> <tbody id="S4.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi_{i,j}^{(a_{i,j})}(x_{n_{i,j}}^{(a_{i,j})})=" class="ltx_Math" display="inline" id="S4.Ex21.m1.11"><semantics id="S4.Ex21.m1.11a"><mrow id="S4.Ex21.m1.11.11" xref="S4.Ex21.m1.11.11.cmml"><mrow id="S4.Ex21.m1.11.11.1" xref="S4.Ex21.m1.11.11.1.cmml"><msubsup id="S4.Ex21.m1.11.11.1.3" xref="S4.Ex21.m1.11.11.1.3.cmml"><mi id="S4.Ex21.m1.11.11.1.3.2.2" mathvariant="normal" xref="S4.Ex21.m1.11.11.1.3.2.2.cmml">Ψ</mi><mrow id="S4.Ex21.m1.2.2.2.4" xref="S4.Ex21.m1.2.2.2.3.cmml"><mi id="S4.Ex21.m1.1.1.1.1" xref="S4.Ex21.m1.1.1.1.1.cmml">i</mi><mo id="S4.Ex21.m1.2.2.2.4.1" xref="S4.Ex21.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex21.m1.2.2.2.2" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) =</annotation></semantics></math></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C_{i,j}B_{i,j}\bigg{[}" class="ltx_math_unparsed" display="inline" id="S4.E33.m1.4"><semantics id="S4.E33.m1.4a"><mrow id="S4.E33.m1.4b"><msub id="S4.E33.m1.4.5"><mi id="S4.E33.m1.4.5.2">C</mi><mrow id="S4.E33.m1.2.2.2.4"><mi id="S4.E33.m1.1.1.1.1">i</mi><mo id="S4.E33.m1.2.2.2.4.1">,</mo><mi id="S4.E33.m1.2.2.2.2">j</mi></mrow></msub><msub id="S4.E33.m1.4.6"><mi id="S4.E33.m1.4.6.2">B</mi><mrow id="S4.E33.m1.4.4.2.4"><mi id="S4.E33.m1.3.3.1.1">i</mi><mo id="S4.E33.m1.4.4.2.4.1">,</mo><mi id="S4.E33.m1.4.4.2.2">j</mi></mrow></msub><mo id="S4.E33.m1.4.7" maxsize="210%" minsize="210%">[</mo></mrow><annotation encoding="application/x-tex" id="S4.E33.m1.4c">\displaystyle C_{i,j}B_{i,j}\bigg{[}</annotation><annotation encoding="application/x-llamapun" id="S4.E33.m1.4d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle g_{+}(a_{i,j})^{\frac{x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}-g_{-}(a% _{i,j})^{\frac{x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}\bigg{]}" class="ltx_math_unparsed" display="inline" id="S4.E33.m2.18"><semantics id="S4.E33.m2.18a"><mrow id="S4.E33.m2.18b"><msub id="S4.E33.m2.18.19"><mi id="S4.E33.m2.18.19.2">g</mi><mo id="S4.E33.m2.18.19.3">+</mo></msub><msup id="S4.E33.m2.18.20"><mrow id="S4.E33.m2.18.20.2"><mo id="S4.E33.m2.18.20.2.1" stretchy="false">(</mo><msub id="S4.E33.m2.18.20.2.2"><mi id="S4.E33.m2.18.20.2.2.2">a</mi><mrow id="S4.E33.m2.2.2.2.4"><mi id="S4.E33.m2.1.1.1.1">i</mi><mo id="S4.E33.m2.2.2.2.4.1">,</mo><mi id="S4.E33.m2.2.2.2.2">j</mi></mrow></msub><mo id="S4.E33.m2.18.20.2.3" stretchy="false">)</mo></mrow><mfrac id="S4.E33.m2.9.9.7"><msubsup id="S4.E33.m2.7.7.5.5.5"><mi id="S4.E33.m2.7.7.5.5.5.7.2">x</mi><msub id="S4.E33.m2.4.4.2.2.2.2.2"><mi id="S4.E33.m2.4.4.2.2.2.2.2.4">n</mi><mrow id="S4.E33.m2.4.4.2.2.2.2.2.2.2.4"><mi id="S4.E33.m2.3.3.1.1.1.1.1.1.1.1">i</mi><mo id="S4.E33.m2.4.4.2.2.2.2.2.2.2.4.1">,</mo><mi id="S4.E33.m2.4.4.2.2.2.2.2.2.2.2">j</mi></mrow></msub><mrow id="S4.E33.m2.7.7.5.5.5.5.3.3"><mo id="S4.E33.m2.7.7.5.5.5.5.3.3.2" stretchy="false">(</mo><msub id="S4.E33.m2.7.7.5.5.5.5.3.3.1"><mi id="S4.E33.m2.7.7.5.5.5.5.3.3.1.2">a</mi><mrow id="S4.E33.m2.6.6.4.4.4.4.2.2.2.4"><mi id="S4.E33.m2.5.5.3.3.3.3.1.1.1.1">i</mi><mo id="S4.E33.m2.6.6.4.4.4.4.2.2.2.4.1">,</mo><mi id="S4.E33.m2.6.6.4.4.4.4.2.2.2.2">j</mi></mrow></msub><mo id="S4.E33.m2.7.7.5.5.5.5.3.3.3" stretchy="false">)</mo></mrow></msubsup><msub id="S4.E33.m2.9.9.7.7.7"><mi id="S4.E33.m2.9.9.7.7.7.4">a</mi><mrow id="S4.E33.m2.9.9.7.7.7.2.2.4"><mi id="S4.E33.m2.8.8.6.6.6.1.1.1">i</mi><mo id="S4.E33.m2.9.9.7.7.7.2.2.4.1">,</mo><mi id="S4.E33.m2.9.9.7.7.7.2.2.2">j</mi></mrow></msub></mfrac></msup><mo id="S4.E33.m2.18.21">−</mo><msub id="S4.E33.m2.18.22"><mi id="S4.E33.m2.18.22.2">g</mi><mo id="S4.E33.m2.18.22.3">−</mo></msub><msup id="S4.E33.m2.18.23"><mrow id="S4.E33.m2.18.23.2"><mo id="S4.E33.m2.18.23.2.1" stretchy="false">(</mo><msub id="S4.E33.m2.18.23.2.2"><mi id="S4.E33.m2.18.23.2.2.2">a</mi><mrow id="S4.E33.m2.11.11.2.4"><mi id="S4.E33.m2.10.10.1.1">i</mi><mo id="S4.E33.m2.11.11.2.4.1">,</mo><mi id="S4.E33.m2.11.11.2.2">j</mi></mrow></msub><mo id="S4.E33.m2.18.23.2.3" stretchy="false">)</mo></mrow><mfrac id="S4.E33.m2.18.18.7"><msubsup id="S4.E33.m2.16.16.5.5.5"><mi id="S4.E33.m2.16.16.5.5.5.7.2">x</mi><msub id="S4.E33.m2.13.13.2.2.2.2.2"><mi id="S4.E33.m2.13.13.2.2.2.2.2.4">n</mi><mrow id="S4.E33.m2.13.13.2.2.2.2.2.2.2.4"><mi id="S4.E33.m2.12.12.1.1.1.1.1.1.1.1">i</mi><mo id="S4.E33.m2.13.13.2.2.2.2.2.2.2.4.1">,</mo><mi id="S4.E33.m2.13.13.2.2.2.2.2.2.2.2">j</mi></mrow></msub><mrow id="S4.E33.m2.16.16.5.5.5.5.3.3"><mo id="S4.E33.m2.16.16.5.5.5.5.3.3.2" stretchy="false">(</mo><msub id="S4.E33.m2.16.16.5.5.5.5.3.3.1"><mi id="S4.E33.m2.16.16.5.5.5.5.3.3.1.2">a</mi><mrow id="S4.E33.m2.15.15.4.4.4.4.2.2.2.4"><mi id="S4.E33.m2.14.14.3.3.3.3.1.1.1.1">i</mi><mo id="S4.E33.m2.15.15.4.4.4.4.2.2.2.4.1">,</mo><mi id="S4.E33.m2.15.15.4.4.4.4.2.2.2.2">j</mi></mrow></msub><mo id="S4.E33.m2.16.16.5.5.5.5.3.3.3" stretchy="false">)</mo></mrow></msubsup><msub id="S4.E33.m2.18.18.7.7.7"><mi id="S4.E33.m2.18.18.7.7.7.4">a</mi><mrow id="S4.E33.m2.18.18.7.7.7.2.2.4"><mi id="S4.E33.m2.17.17.6.6.6.1.1.1">i</mi><mo id="S4.E33.m2.18.18.7.7.7.2.2.4.1">,</mo><mi id="S4.E33.m2.18.18.7.7.7.2.2.2">j</mi></mrow></msub></mfrac></msup><mo id="S4.E33.m2.18.24" maxsize="210%" minsize="210%">]</mo></mrow><annotation encoding="application/x-tex" id="S4.E33.m2.18c">\displaystyle g_{+}(a_{i,j})^{\frac{x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}-g_{-}(a% _{i,j})^{\frac{x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}\bigg{]}</annotation><annotation encoding="application/x-llamapun" id="S4.E33.m2.18d">italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT - italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(33)</span></td> </tr></tbody> <tbody id="S4.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle+C_{i,j}A_{i,j}\bigg{[}" class="ltx_math_unparsed" display="inline" id="S4.Ex22.m1.4"><semantics id="S4.Ex22.m1.4a"><mrow id="S4.Ex22.m1.4b"><mo id="S4.Ex22.m1.4.5">+</mo><msub id="S4.Ex22.m1.4.6"><mi id="S4.Ex22.m1.4.6.2">C</mi><mrow id="S4.Ex22.m1.2.2.2.4"><mi id="S4.Ex22.m1.1.1.1.1">i</mi><mo id="S4.Ex22.m1.2.2.2.4.1">,</mo><mi id="S4.Ex22.m1.2.2.2.2">j</mi></mrow></msub><msub id="S4.Ex22.m1.4.7"><mi id="S4.Ex22.m1.4.7.2">A</mi><mrow id="S4.Ex22.m1.4.4.2.4"><mi id="S4.Ex22.m1.3.3.1.1">i</mi><mo id="S4.Ex22.m1.4.4.2.4.1">,</mo><mi id="S4.Ex22.m1.4.4.2.2">j</mi></mrow></msub><mo id="S4.Ex22.m1.4.8" maxsize="210%" minsize="210%">[</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex22.m1.4c">\displaystyle+C_{i,j}A_{i,j}\bigg{[}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex22.m1.4d">+ italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle g_{+}(a_{i,j})^{\frac{L_{i,j}-x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}% -g_{-}(a_{i,j})^{\frac{L_{i,j}-x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}\bigg{]}," class="ltx_math_unparsed" display="inline" id="S4.Ex22.m2.22"><semantics id="S4.Ex22.m2.22a"><mrow id="S4.Ex22.m2.22b"><msub id="S4.Ex22.m2.22.23"><mi id="S4.Ex22.m2.22.23.2">g</mi><mo id="S4.Ex22.m2.22.23.3">+</mo></msub><msup id="S4.Ex22.m2.22.24"><mrow id="S4.Ex22.m2.22.24.2"><mo id="S4.Ex22.m2.22.24.2.1" stretchy="false">(</mo><msub id="S4.Ex22.m2.22.24.2.2"><mi id="S4.Ex22.m2.22.24.2.2.2">a</mi><mrow id="S4.Ex22.m2.2.2.2.4"><mi id="S4.Ex22.m2.1.1.1.1">i</mi><mo id="S4.Ex22.m2.2.2.2.4.1">,</mo><mi id="S4.Ex22.m2.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex22.m2.22.24.2.3" stretchy="false">)</mo></mrow><mfrac id="S4.Ex22.m2.11.11.9"><mrow id="S4.Ex22.m2.9.9.7.7.7"><msub id="S4.Ex22.m2.9.9.7.7.7.9"><mi id="S4.Ex22.m2.9.9.7.7.7.9.2">L</mi><mrow id="S4.Ex22.m2.4.4.2.2.2.2.2.4"><mi id="S4.Ex22.m2.3.3.1.1.1.1.1.1">i</mi><mo id="S4.Ex22.m2.4.4.2.2.2.2.2.4.1">,</mo><mi id="S4.Ex22.m2.4.4.2.2.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex22.m2.9.9.7.7.7.8">−</mo><msubsup id="S4.Ex22.m2.9.9.7.7.7.10"><mi id="S4.Ex22.m2.9.9.7.7.7.10.2.2">x</mi><msub id="S4.Ex22.m2.6.6.4.4.4.4.2"><mi id="S4.Ex22.m2.6.6.4.4.4.4.2.4">n</mi><mrow id="S4.Ex22.m2.6.6.4.4.4.4.2.2.2.4"><mi id="S4.Ex22.m2.5.5.3.3.3.3.1.1.1.1">i</mi><mo id="S4.Ex22.m2.6.6.4.4.4.4.2.2.2.4.1">,</mo><mi id="S4.Ex22.m2.6.6.4.4.4.4.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex22.m2.9.9.7.7.7.7.3.3"><mo id="S4.Ex22.m2.9.9.7.7.7.7.3.3.2" stretchy="false">(</mo><msub id="S4.Ex22.m2.9.9.7.7.7.7.3.3.1"><mi id="S4.Ex22.m2.9.9.7.7.7.7.3.3.1.2">a</mi><mrow id="S4.Ex22.m2.8.8.6.6.6.6.2.2.2.4"><mi id="S4.Ex22.m2.7.7.5.5.5.5.1.1.1.1">i</mi><mo id="S4.Ex22.m2.8.8.6.6.6.6.2.2.2.4.1">,</mo><mi id="S4.Ex22.m2.8.8.6.6.6.6.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex22.m2.9.9.7.7.7.7.3.3.3" stretchy="false">)</mo></mrow></msubsup></mrow><msub id="S4.Ex22.m2.11.11.9.9.9"><mi id="S4.Ex22.m2.11.11.9.9.9.4">a</mi><mrow id="S4.Ex22.m2.11.11.9.9.9.2.2.4"><mi id="S4.Ex22.m2.10.10.8.8.8.1.1.1">i</mi><mo id="S4.Ex22.m2.11.11.9.9.9.2.2.4.1">,</mo><mi id="S4.Ex22.m2.11.11.9.9.9.2.2.2">j</mi></mrow></msub></mfrac></msup><mo id="S4.Ex22.m2.22.25">−</mo><msub id="S4.Ex22.m2.22.26"><mi id="S4.Ex22.m2.22.26.2">g</mi><mo id="S4.Ex22.m2.22.26.3">−</mo></msub><msup id="S4.Ex22.m2.22.27"><mrow id="S4.Ex22.m2.22.27.2"><mo id="S4.Ex22.m2.22.27.2.1" stretchy="false">(</mo><msub id="S4.Ex22.m2.22.27.2.2"><mi id="S4.Ex22.m2.22.27.2.2.2">a</mi><mrow id="S4.Ex22.m2.13.13.2.4"><mi id="S4.Ex22.m2.12.12.1.1">i</mi><mo id="S4.Ex22.m2.13.13.2.4.1">,</mo><mi id="S4.Ex22.m2.13.13.2.2">j</mi></mrow></msub><mo id="S4.Ex22.m2.22.27.2.3" stretchy="false">)</mo></mrow><mfrac id="S4.Ex22.m2.22.22.9"><mrow id="S4.Ex22.m2.20.20.7.7.7"><msub id="S4.Ex22.m2.20.20.7.7.7.9"><mi id="S4.Ex22.m2.20.20.7.7.7.9.2">L</mi><mrow id="S4.Ex22.m2.15.15.2.2.2.2.2.4"><mi id="S4.Ex22.m2.14.14.1.1.1.1.1.1">i</mi><mo id="S4.Ex22.m2.15.15.2.2.2.2.2.4.1">,</mo><mi id="S4.Ex22.m2.15.15.2.2.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex22.m2.20.20.7.7.7.8">−</mo><msubsup id="S4.Ex22.m2.20.20.7.7.7.10"><mi id="S4.Ex22.m2.20.20.7.7.7.10.2.2">x</mi><msub id="S4.Ex22.m2.17.17.4.4.4.4.2"><mi id="S4.Ex22.m2.17.17.4.4.4.4.2.4">n</mi><mrow id="S4.Ex22.m2.17.17.4.4.4.4.2.2.2.4"><mi id="S4.Ex22.m2.16.16.3.3.3.3.1.1.1.1">i</mi><mo id="S4.Ex22.m2.17.17.4.4.4.4.2.2.2.4.1">,</mo><mi id="S4.Ex22.m2.17.17.4.4.4.4.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex22.m2.20.20.7.7.7.7.3.3"><mo id="S4.Ex22.m2.20.20.7.7.7.7.3.3.2" stretchy="false">(</mo><msub id="S4.Ex22.m2.20.20.7.7.7.7.3.3.1"><mi id="S4.Ex22.m2.20.20.7.7.7.7.3.3.1.2">a</mi><mrow id="S4.Ex22.m2.19.19.6.6.6.6.2.2.2.4"><mi id="S4.Ex22.m2.18.18.5.5.5.5.1.1.1.1">i</mi><mo id="S4.Ex22.m2.19.19.6.6.6.6.2.2.2.4.1">,</mo><mi id="S4.Ex22.m2.19.19.6.6.6.6.2.2.2.2">j</mi></mrow></msub><mo id="S4.Ex22.m2.20.20.7.7.7.7.3.3.3" stretchy="false">)</mo></mrow></msubsup></mrow><msub id="S4.Ex22.m2.22.22.9.9.9"><mi id="S4.Ex22.m2.22.22.9.9.9.4">a</mi><mrow id="S4.Ex22.m2.22.22.9.9.9.2.2.4"><mi id="S4.Ex22.m2.21.21.8.8.8.1.1.1">i</mi><mo id="S4.Ex22.m2.22.22.9.9.9.2.2.4.1">,</mo><mi id="S4.Ex22.m2.22.22.9.9.9.2.2.2">j</mi></mrow></msub></mfrac></msup><mo id="S4.Ex22.m2.22.28" maxsize="210%" minsize="210%">]</mo><mo id="S4.Ex22.m2.22.29">,</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex22.m2.22c">\displaystyle g_{+}(a_{i,j})^{\frac{L_{i,j}-x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}% -g_{-}(a_{i,j})^{\frac{L_{i,j}-x_{n_{i,j}}^{(a_{i,j})}}{a_{i,j}}}\bigg{]},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex22.m2.22d">italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT - italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p10.1">for <math alttext="i<j" class="ltx_Math" display="inline" id="S4.SS1.p10.1.m1.1"><semantics id="S4.SS1.p10.1.m1.1a"><mrow id="S4.SS1.p10.1.m1.1.1" xref="S4.SS1.p10.1.m1.1.1.cmml"><mi id="S4.SS1.p10.1.m1.1.1.2" xref="S4.SS1.p10.1.m1.1.1.2.cmml">i</mi><mo id="S4.SS1.p10.1.m1.1.1.1" xref="S4.SS1.p10.1.m1.1.1.1.cmml"><</mo><mi id="S4.SS1.p10.1.m1.1.1.3" xref="S4.SS1.p10.1.m1.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p10.1.m1.1b"><apply id="S4.SS1.p10.1.m1.1.1.cmml" xref="S4.SS1.p10.1.m1.1.1"><lt id="S4.SS1.p10.1.m1.1.1.1.cmml" xref="S4.SS1.p10.1.m1.1.1.1"></lt><ci id="S4.SS1.p10.1.m1.1.1.2.cmml" xref="S4.SS1.p10.1.m1.1.1.2">𝑖</ci><ci id="S4.SS1.p10.1.m1.1.1.3.cmml" xref="S4.SS1.p10.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p10.1.m1.1c">i<j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p10.1.m1.1d">italic_i < italic_j</annotation></semantics></math>, where</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{\pm}(a_{i,j})=1+\frac{-k^{2}a^{2}_{i,j}\pm ka_{i,j}\sqrt{k^{2}a^{2}_{i,j}-4% }}{2}." class="ltx_Math" display="block" id="S4.Ex23.m1.9"><semantics id="S4.Ex23.m1.9a"><mrow id="S4.Ex23.m1.9.9.1" xref="S4.Ex23.m1.9.9.1.1.cmml"><mrow id="S4.Ex23.m1.9.9.1.1" xref="S4.Ex23.m1.9.9.1.1.cmml"><mrow id="S4.Ex23.m1.9.9.1.1.1" xref="S4.Ex23.m1.9.9.1.1.1.cmml"><msub id="S4.Ex23.m1.9.9.1.1.1.3" xref="S4.Ex23.m1.9.9.1.1.1.3.cmml"><mi id="S4.Ex23.m1.9.9.1.1.1.3.2" xref="S4.Ex23.m1.9.9.1.1.1.3.2.cmml">g</mi><mo id="S4.Ex23.m1.9.9.1.1.1.3.3" xref="S4.Ex23.m1.9.9.1.1.1.3.3.cmml">±</mo></msub><mo id="S4.Ex23.m1.9.9.1.1.1.2" xref="S4.Ex23.m1.9.9.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex23.m1.9.9.1.1.1.1.1" 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id="S4.Ex23.m1.8.8.6.6.2" xref="S4.Ex23.m1.8.8.6.6.2.cmml"><mrow id="S4.Ex23.m1.8.8.6.6.2.4" xref="S4.Ex23.m1.8.8.6.6.2.4.cmml"><msup id="S4.Ex23.m1.8.8.6.6.2.4.2" xref="S4.Ex23.m1.8.8.6.6.2.4.2.cmml"><mi id="S4.Ex23.m1.8.8.6.6.2.4.2.2" xref="S4.Ex23.m1.8.8.6.6.2.4.2.2.cmml">k</mi><mn id="S4.Ex23.m1.8.8.6.6.2.4.2.3" xref="S4.Ex23.m1.8.8.6.6.2.4.2.3.cmml">2</mn></msup><mo id="S4.Ex23.m1.8.8.6.6.2.4.1" xref="S4.Ex23.m1.8.8.6.6.2.4.1.cmml">⁢</mo><msubsup id="S4.Ex23.m1.8.8.6.6.2.4.3" xref="S4.Ex23.m1.8.8.6.6.2.4.3.cmml"><mi id="S4.Ex23.m1.8.8.6.6.2.4.3.2.2" xref="S4.Ex23.m1.8.8.6.6.2.4.3.2.2.cmml">a</mi><mrow id="S4.Ex23.m1.8.8.6.6.2.2.2.4" xref="S4.Ex23.m1.8.8.6.6.2.2.2.3.cmml"><mi id="S4.Ex23.m1.7.7.5.5.1.1.1.1" xref="S4.Ex23.m1.7.7.5.5.1.1.1.1.cmml">i</mi><mo id="S4.Ex23.m1.8.8.6.6.2.2.2.4.1" xref="S4.Ex23.m1.8.8.6.6.2.2.2.3.cmml">,</mo><mi id="S4.Ex23.m1.8.8.6.6.2.2.2.2" xref="S4.Ex23.m1.8.8.6.6.2.2.2.2.cmml">j</mi></mrow><mn id="S4.Ex23.m1.8.8.6.6.2.4.3.2.3" xref="S4.Ex23.m1.8.8.6.6.2.4.3.2.3.cmml">2</mn></msubsup></mrow><mo id="S4.Ex23.m1.8.8.6.6.2.3" xref="S4.Ex23.m1.8.8.6.6.2.3.cmml">−</mo><mn id="S4.Ex23.m1.8.8.6.6.2.5" xref="S4.Ex23.m1.8.8.6.6.2.5.cmml">4</mn></mrow></msqrt></mrow></mrow><mn id="S4.Ex23.m1.8.8.8" xref="S4.Ex23.m1.8.8.8.cmml">2</mn></mfrac></mrow></mrow><mo id="S4.Ex23.m1.9.9.1.2" lspace="0em" xref="S4.Ex23.m1.9.9.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex23.m1.9b"><apply id="S4.Ex23.m1.9.9.1.1.cmml" xref="S4.Ex23.m1.9.9.1"><eq id="S4.Ex23.m1.9.9.1.1.2.cmml" xref="S4.Ex23.m1.9.9.1.1.2"></eq><apply id="S4.Ex23.m1.9.9.1.1.1.cmml" xref="S4.Ex23.m1.9.9.1.1.1"><times id="S4.Ex23.m1.9.9.1.1.1.2.cmml" xref="S4.Ex23.m1.9.9.1.1.1.2"></times><apply id="S4.Ex23.m1.9.9.1.1.1.3.cmml" xref="S4.Ex23.m1.9.9.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex23.m1.9.9.1.1.1.3.1.cmml" xref="S4.Ex23.m1.9.9.1.1.1.3">subscript</csymbol><ci id="S4.Ex23.m1.9.9.1.1.1.3.2.cmml" 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xref="S4.Ex23.m1.8.8.6.6.2.2.2.2">𝑗</ci></list></apply></apply><cn id="S4.Ex23.m1.8.8.6.6.2.5.cmml" type="integer" xref="S4.Ex23.m1.8.8.6.6.2.5">4</cn></apply></apply></apply></apply><cn id="S4.Ex23.m1.8.8.8.cmml" type="integer" xref="S4.Ex23.m1.8.8.8">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex23.m1.9c">g_{\pm}(a_{i,j})=1+\frac{-k^{2}a^{2}_{i,j}\pm ka_{i,j}\sqrt{k^{2}a^{2}_{i,j}-4% }}{2}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex23.m1.9d">italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) = 1 + divide start_ARG - italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ± italic_k italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - 4 end_ARG end_ARG start_ARG 2 end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p11"> <p class="ltx_p" id="S4.SS1.p11.2">These functions are the exact solution of the discrete Schrödinger equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.E28" title="In 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">28</span></a>). Continuous limit of the solution can be obtained by considering the limit at <math alttext="a_{i,j}\to 0" class="ltx_Math" display="inline" id="S4.SS1.p11.1.m1.2"><semantics id="S4.SS1.p11.1.m1.2a"><mrow id="S4.SS1.p11.1.m1.2.3" xref="S4.SS1.p11.1.m1.2.3.cmml"><msub id="S4.SS1.p11.1.m1.2.3.2" xref="S4.SS1.p11.1.m1.2.3.2.cmml"><mi id="S4.SS1.p11.1.m1.2.3.2.2" xref="S4.SS1.p11.1.m1.2.3.2.2.cmml">a</mi><mrow id="S4.SS1.p11.1.m1.2.2.2.4" xref="S4.SS1.p11.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p11.1.m1.1.1.1.1" xref="S4.SS1.p11.1.m1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p11.1.m1.2.2.2.4.1" xref="S4.SS1.p11.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p11.1.m1.2.2.2.2" xref="S4.SS1.p11.1.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p11.1.m1.2.3.1" stretchy="false" xref="S4.SS1.p11.1.m1.2.3.1.cmml">→</mo><mn id="S4.SS1.p11.1.m1.2.3.3" xref="S4.SS1.p11.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p11.1.m1.2b"><apply id="S4.SS1.p11.1.m1.2.3.cmml" xref="S4.SS1.p11.1.m1.2.3"><ci id="S4.SS1.p11.1.m1.2.3.1.cmml" xref="S4.SS1.p11.1.m1.2.3.1">→</ci><apply id="S4.SS1.p11.1.m1.2.3.2.cmml" xref="S4.SS1.p11.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p11.1.m1.2.3.2.1.cmml" xref="S4.SS1.p11.1.m1.2.3.2">subscript</csymbol><ci id="S4.SS1.p11.1.m1.2.3.2.2.cmml" xref="S4.SS1.p11.1.m1.2.3.2.2">𝑎</ci><list id="S4.SS1.p11.1.m1.2.2.2.3.cmml" xref="S4.SS1.p11.1.m1.2.2.2.4"><ci id="S4.SS1.p11.1.m1.1.1.1.1.cmml" xref="S4.SS1.p11.1.m1.1.1.1.1">𝑖</ci><ci id="S4.SS1.p11.1.m1.2.2.2.2.cmml" xref="S4.SS1.p11.1.m1.2.2.2.2">𝑗</ci></list></apply><cn id="S4.SS1.p11.1.m1.2.3.3.cmml" type="integer" xref="S4.SS1.p11.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p11.1.m1.2c">a_{i,j}\to 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p11.1.m1.2d">italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT → 0</annotation></semantics></math> for fixed point <math alttext="x_{i,j}^{(a_{i,j})}=\alpha_{i,j}" class="ltx_Math" display="inline" id="S4.SS1.p11.2.m2.7"><semantics id="S4.SS1.p11.2.m2.7a"><mrow id="S4.SS1.p11.2.m2.7.8" xref="S4.SS1.p11.2.m2.7.8.cmml"><msubsup id="S4.SS1.p11.2.m2.7.8.2" xref="S4.SS1.p11.2.m2.7.8.2.cmml"><mi id="S4.SS1.p11.2.m2.7.8.2.2.2" xref="S4.SS1.p11.2.m2.7.8.2.2.2.cmml">x</mi><mrow id="S4.SS1.p11.2.m2.2.2.2.4" xref="S4.SS1.p11.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p11.2.m2.1.1.1.1" xref="S4.SS1.p11.2.m2.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p11.2.m2.2.2.2.4.1" xref="S4.SS1.p11.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p11.2.m2.2.2.2.2" xref="S4.SS1.p11.2.m2.2.2.2.2.cmml">j</mi></mrow><mrow id="S4.SS1.p11.2.m2.5.5.3.3" xref="S4.SS1.p11.2.m2.5.5.3.3.1.cmml"><mo id="S4.SS1.p11.2.m2.5.5.3.3.2" stretchy="false" xref="S4.SS1.p11.2.m2.5.5.3.3.1.cmml">(</mo><msub id="S4.SS1.p11.2.m2.5.5.3.3.1" xref="S4.SS1.p11.2.m2.5.5.3.3.1.cmml"><mi id="S4.SS1.p11.2.m2.5.5.3.3.1.2" xref="S4.SS1.p11.2.m2.5.5.3.3.1.2.cmml">a</mi><mrow id="S4.SS1.p11.2.m2.4.4.2.2.2.4" xref="S4.SS1.p11.2.m2.4.4.2.2.2.3.cmml"><mi id="S4.SS1.p11.2.m2.3.3.1.1.1.1" xref="S4.SS1.p11.2.m2.3.3.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p11.2.m2.4.4.2.2.2.4.1" xref="S4.SS1.p11.2.m2.4.4.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p11.2.m2.4.4.2.2.2.2" xref="S4.SS1.p11.2.m2.4.4.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p11.2.m2.5.5.3.3.3" stretchy="false" xref="S4.SS1.p11.2.m2.5.5.3.3.1.cmml">)</mo></mrow></msubsup><mo id="S4.SS1.p11.2.m2.7.8.1" xref="S4.SS1.p11.2.m2.7.8.1.cmml">=</mo><msub id="S4.SS1.p11.2.m2.7.8.3" xref="S4.SS1.p11.2.m2.7.8.3.cmml"><mi id="S4.SS1.p11.2.m2.7.8.3.2" xref="S4.SS1.p11.2.m2.7.8.3.2.cmml">α</mi><mrow id="S4.SS1.p11.2.m2.7.7.2.4" xref="S4.SS1.p11.2.m2.7.7.2.3.cmml"><mi id="S4.SS1.p11.2.m2.6.6.1.1" xref="S4.SS1.p11.2.m2.6.6.1.1.cmml">i</mi><mo id="S4.SS1.p11.2.m2.7.7.2.4.1" xref="S4.SS1.p11.2.m2.7.7.2.3.cmml">,</mo><mi id="S4.SS1.p11.2.m2.7.7.2.2" xref="S4.SS1.p11.2.m2.7.7.2.2.cmml">j</mi></mrow></msub></mrow><annotation-xml 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id="S4.SS1.p11.2.m2.7c">x_{i,j}^{(a_{i,j})}=\alpha_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p11.2.m2.7d">italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT = italic_α start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, as it is shown in the section <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2" title="2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a>, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx27"> <tbody id="S4.Ex24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\lim_{a_{i,j}\to 0}\Psi_{i,j}^{(a_{i,j})}(\alpha_{i,j})=" class="ltx_Math" display="inline" id="S4.Ex24.m1.10"><semantics id="S4.Ex24.m1.10a"><mrow id="S4.Ex24.m1.10.10" xref="S4.Ex24.m1.10.10.cmml"><mrow id="S4.Ex24.m1.10.10.1" xref="S4.Ex24.m1.10.10.1.cmml"><munder id="S4.Ex24.m1.10.10.1.2" xref="S4.Ex24.m1.10.10.1.2.cmml"><mo id="S4.Ex24.m1.10.10.1.2.2" movablelimits="false" xref="S4.Ex24.m1.10.10.1.2.2.cmml">lim</mo><mrow id="S4.Ex24.m1.2.2.2" xref="S4.Ex24.m1.2.2.2.cmml"><msub id="S4.Ex24.m1.2.2.2.4" xref="S4.Ex24.m1.2.2.2.4.cmml"><mi id="S4.Ex24.m1.2.2.2.4.2" xref="S4.Ex24.m1.2.2.2.4.2.cmml">a</mi><mrow id="S4.Ex24.m1.2.2.2.2.2.4" xref="S4.Ex24.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex24.m1.1.1.1.1.1.1" xref="S4.Ex24.m1.1.1.1.1.1.1.cmml">i</mi><mo id="S4.Ex24.m1.2.2.2.2.2.4.1" xref="S4.Ex24.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.Ex24.m1.2.2.2.2.2.2" xref="S4.Ex24.m1.2.2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.Ex24.m1.2.2.2.3" stretchy="false" xref="S4.Ex24.m1.2.2.2.3.cmml">→</mo><mn id="S4.Ex24.m1.2.2.2.5" xref="S4.Ex24.m1.2.2.2.5.cmml">0</mn></mrow></munder><mrow id="S4.Ex24.m1.10.10.1.1" xref="S4.Ex24.m1.10.10.1.1.cmml"><msubsup id="S4.Ex24.m1.10.10.1.1.3" xref="S4.Ex24.m1.10.10.1.1.3.cmml"><mi id="S4.Ex24.m1.10.10.1.1.3.2.2" mathvariant="normal" xref="S4.Ex24.m1.10.10.1.1.3.2.2.cmml">Ψ</mi><mrow id="S4.Ex24.m1.4.4.2.4" xref="S4.Ex24.m1.4.4.2.3.cmml"><mi id="S4.Ex24.m1.3.3.1.1" xref="S4.Ex24.m1.3.3.1.1.cmml">i</mi><mo id="S4.Ex24.m1.4.4.2.4.1" xref="S4.Ex24.m1.4.4.2.3.cmml">,</mo><mi id="S4.Ex24.m1.4.4.2.2" xref="S4.Ex24.m1.4.4.2.2.cmml">j</mi></mrow><mrow id="S4.Ex24.m1.7.7.3.3" xref="S4.Ex24.m1.7.7.3.3.1.cmml"><mo id="S4.Ex24.m1.7.7.3.3.2" stretchy="false" xref="S4.Ex24.m1.7.7.3.3.1.cmml">(</mo><msub id="S4.Ex24.m1.7.7.3.3.1" xref="S4.Ex24.m1.7.7.3.3.1.cmml"><mi id="S4.Ex24.m1.7.7.3.3.1.2" xref="S4.Ex24.m1.7.7.3.3.1.2.cmml">a</mi><mrow id="S4.Ex24.m1.6.6.2.2.2.4" xref="S4.Ex24.m1.6.6.2.2.2.3.cmml"><mi id="S4.Ex24.m1.5.5.1.1.1.1" xref="S4.Ex24.m1.5.5.1.1.1.1.cmml">i</mi><mo id="S4.Ex24.m1.6.6.2.2.2.4.1" xref="S4.Ex24.m1.6.6.2.2.2.3.cmml">,</mo><mi id="S4.Ex24.m1.6.6.2.2.2.2" xref="S4.Ex24.m1.6.6.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.Ex24.m1.7.7.3.3.3" stretchy="false" xref="S4.Ex24.m1.7.7.3.3.1.cmml">)</mo></mrow></msubsup><mo id="S4.Ex24.m1.10.10.1.1.2" xref="S4.Ex24.m1.10.10.1.1.2.cmml">⁢</mo><mrow id="S4.Ex24.m1.10.10.1.1.1.1" xref="S4.Ex24.m1.10.10.1.1.1.1.1.cmml"><mo id="S4.Ex24.m1.10.10.1.1.1.1.2" stretchy="false" xref="S4.Ex24.m1.10.10.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex24.m1.10.10.1.1.1.1.1" xref="S4.Ex24.m1.10.10.1.1.1.1.1.cmml"><mi id="S4.Ex24.m1.10.10.1.1.1.1.1.2" xref="S4.Ex24.m1.10.10.1.1.1.1.1.2.cmml">α</mi><mrow id="S4.Ex24.m1.9.9.2.4" xref="S4.Ex24.m1.9.9.2.3.cmml"><mi id="S4.Ex24.m1.8.8.1.1" xref="S4.Ex24.m1.8.8.1.1.cmml">i</mi><mo id="S4.Ex24.m1.9.9.2.4.1" xref="S4.Ex24.m1.9.9.2.3.cmml">,</mo><mi id="S4.Ex24.m1.9.9.2.2" xref="S4.Ex24.m1.9.9.2.2.cmml">j</mi></mrow></msub><mo id="S4.Ex24.m1.10.10.1.1.1.1.3" stretchy="false" xref="S4.Ex24.m1.10.10.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex24.m1.10.10.2" xref="S4.Ex24.m1.10.10.2.cmml">=</mo><mi id="S4.Ex24.m1.10.10.3" xref="S4.Ex24.m1.10.10.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex24.m1.10b"><apply id="S4.Ex24.m1.10.10.cmml" 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xref="S4.Ex24.m1.10.10.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex24.m1.10c">\displaystyle\lim_{a_{i,j}\to 0}\Psi_{i,j}^{(a_{i,j})}(\alpha_{i,j})=</annotation><annotation encoding="application/x-llamapun" id="S4.Ex24.m1.10d">roman_lim start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT → 0 end_POSTSUBSCRIPT roman_Ψ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_α start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) =</annotation></semantics></math></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 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encoding="MathML-Content" id="S4.E34.m1.4b"><apply id="S4.E34.m1.4.5.cmml" xref="S4.E34.m1.4.5"><times id="S4.E34.m1.4.5.1.cmml" xref="S4.E34.m1.4.5.1"></times><apply id="S4.E34.m1.4.5.2.cmml" xref="S4.E34.m1.4.5.2"><csymbol cd="ambiguous" id="S4.E34.m1.4.5.2.1.cmml" xref="S4.E34.m1.4.5.2">subscript</csymbol><ci id="S4.E34.m1.4.5.2.2.cmml" xref="S4.E34.m1.4.5.2.2">𝐶</ci><list id="S4.E34.m1.2.2.2.3.cmml" xref="S4.E34.m1.2.2.2.4"><ci id="S4.E34.m1.1.1.1.1.cmml" xref="S4.E34.m1.1.1.1.1">𝑖</ci><ci id="S4.E34.m1.2.2.2.2.cmml" xref="S4.E34.m1.2.2.2.2">𝑗</ci></list></apply><apply id="S4.E34.m1.4.5.3.cmml" xref="S4.E34.m1.4.5.3"><csymbol cd="ambiguous" id="S4.E34.m1.4.5.3.1.cmml" xref="S4.E34.m1.4.5.3">subscript</csymbol><ci id="S4.E34.m1.4.5.3.2.cmml" xref="S4.E34.m1.4.5.3.2">𝐵</ci><list id="S4.E34.m1.4.4.2.3.cmml" xref="S4.E34.m1.4.4.2.4"><ci id="S4.E34.m1.3.3.1.1.cmml" xref="S4.E34.m1.3.3.1.1">𝑖</ci><ci id="S4.E34.m1.4.4.2.2.cmml" xref="S4.E34.m1.4.4.2.2">𝑗</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E34.m1.4c">\displaystyle C_{i,j}B_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S4.E34.m1.4d">italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\left[e^{ik\alpha_{i,j}}-e^{-ik\alpha_{i,j}}\right]" class="ltx_Math" display="inline" id="S4.E34.m2.5"><semantics id="S4.E34.m2.5a"><mrow id="S4.E34.m2.5.5.1" xref="S4.E34.m2.5.5.2.cmml"><mo id="S4.E34.m2.5.5.1.2" xref="S4.E34.m2.5.5.2.1.cmml">[</mo><mrow id="S4.E34.m2.5.5.1.1" xref="S4.E34.m2.5.5.1.1.cmml"><msup id="S4.E34.m2.5.5.1.1.2" xref="S4.E34.m2.5.5.1.1.2.cmml"><mi id="S4.E34.m2.5.5.1.1.2.2" xref="S4.E34.m2.5.5.1.1.2.2.cmml">e</mi><mrow id="S4.E34.m2.2.2.2" xref="S4.E34.m2.2.2.2.cmml"><mi 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xref="S4.E34.m2.4.4.2.4.4">subscript</csymbol><ci id="S4.E34.m2.4.4.2.4.4.2.cmml" xref="S4.E34.m2.4.4.2.4.4.2">𝛼</ci><list id="S4.E34.m2.4.4.2.2.2.3.cmml" xref="S4.E34.m2.4.4.2.2.2.4"><ci id="S4.E34.m2.3.3.1.1.1.1.cmml" xref="S4.E34.m2.3.3.1.1.1.1">𝑖</ci><ci id="S4.E34.m2.4.4.2.2.2.2.cmml" xref="S4.E34.m2.4.4.2.2.2.2">𝑗</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E34.m2.5c">\displaystyle\left[e^{ik\alpha_{i,j}}-e^{-ik\alpha_{i,j}}\right]</annotation><annotation encoding="application/x-llamapun" id="S4.E34.m2.5d">[ italic_e start_POSTSUPERSCRIPT italic_i italic_k italic_α start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT - italic_i italic_k italic_α start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(34)</span></td> </tr></tbody> <tbody id="S4.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle+C_{i,j}A_{i,j}" class="ltx_Math" display="inline" id="S4.Ex25.m1.4"><semantics id="S4.Ex25.m1.4a"><mrow id="S4.Ex25.m1.4.5" xref="S4.Ex25.m1.4.5.cmml"><mo id="S4.Ex25.m1.4.5a" xref="S4.Ex25.m1.4.5.cmml">+</mo><mrow id="S4.Ex25.m1.4.5.2" xref="S4.Ex25.m1.4.5.2.cmml"><msub id="S4.Ex25.m1.4.5.2.2" xref="S4.Ex25.m1.4.5.2.2.cmml"><mi id="S4.Ex25.m1.4.5.2.2.2" xref="S4.Ex25.m1.4.5.2.2.2.cmml">C</mi><mrow id="S4.Ex25.m1.2.2.2.4" xref="S4.Ex25.m1.2.2.2.3.cmml"><mi id="S4.Ex25.m1.1.1.1.1" xref="S4.Ex25.m1.1.1.1.1.cmml">i</mi><mo id="S4.Ex25.m1.2.2.2.4.1" xref="S4.Ex25.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex25.m1.2.2.2.2" 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id="S4.Ex25.m2.11c">\displaystyle\left[e^{ik\left(L_{i,j}-\alpha_{i,j}\right)}-e^{-ik\left(L_{i,j}% -\alpha_{i,j}\right)}\right].</annotation><annotation encoding="application/x-llamapun" id="S4.Ex25.m2.11d">[ italic_e start_POSTSUPERSCRIPT italic_i italic_k ( italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT - italic_i italic_k ( italic_L start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p11.3">The right-hand side of Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.Ex24" title="4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">4.1</span></a>) is analogue of the solution of continuous Schrödinger equation on a continuous graph in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E24" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">24</span></a>).</p> </div> <figure class="ltx_table" id="S4.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 2: </span>The first five nonzero eigenvalues, <math alttext="k" class="ltx_Math" display="inline" id="S4.T2.2.m1.1"><semantics id="S4.T2.2.m1.1b"><mi id="S4.T2.2.m1.1.1" xref="S4.T2.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.T2.2.m1.1c"><ci id="S4.T2.2.m1.1.1.cmml" xref="S4.T2.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.2.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="S4.T2.2.m1.1e">italic_k</annotation></semantics></math> compared with the continuous case on the star graph.</figcaption> <table class="ltx_tabular 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id="S4.T2.3.1.1.m1.1.1.2.cmml" xref="S4.T2.3.1.1.m1.1.1.2">𝑎</ci><cn id="S4.T2.3.1.1.m1.1.1.3.cmml" type="float" xref="S4.T2.3.1.1.m1.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.3.1.1.m1.1c">a=0.1</annotation><annotation encoding="application/x-llamapun" id="S4.T2.3.1.1.m1.1d">italic_a = 0.1</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S4.T2.4.2.2"><math alttext="a=0.01" class="ltx_Math" display="inline" id="S4.T2.4.2.2.m1.1"><semantics id="S4.T2.4.2.2.m1.1a"><mrow id="S4.T2.4.2.2.m1.1.1" xref="S4.T2.4.2.2.m1.1.1.cmml"><mi id="S4.T2.4.2.2.m1.1.1.2" xref="S4.T2.4.2.2.m1.1.1.2.cmml">a</mi><mo id="S4.T2.4.2.2.m1.1.1.1" xref="S4.T2.4.2.2.m1.1.1.1.cmml">=</mo><mn id="S4.T2.4.2.2.m1.1.1.3" xref="S4.T2.4.2.2.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.4.2.2.m1.1b"><apply id="S4.T2.4.2.2.m1.1.1.cmml" xref="S4.T2.4.2.2.m1.1.1"><eq id="S4.T2.4.2.2.m1.1.1.1.cmml" xref="S4.T2.4.2.2.m1.1.1.1"></eq><ci id="S4.T2.4.2.2.m1.1.1.2.cmml" xref="S4.T2.4.2.2.m1.1.1.2">𝑎</ci><cn id="S4.T2.4.2.2.m1.1.1.3.cmml" type="float" xref="S4.T2.4.2.2.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.4.2.2.m1.1c">a=0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T2.4.2.2.m1.1d">italic_a = 0.01</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T2.5.3.3"><math alttext="a=0.005" class="ltx_Math" display="inline" id="S4.T2.5.3.3.m1.1"><semantics id="S4.T2.5.3.3.m1.1a"><mrow id="S4.T2.5.3.3.m1.1.1" xref="S4.T2.5.3.3.m1.1.1.cmml"><mi id="S4.T2.5.3.3.m1.1.1.2" xref="S4.T2.5.3.3.m1.1.1.2.cmml">a</mi><mo id="S4.T2.5.3.3.m1.1.1.1" xref="S4.T2.5.3.3.m1.1.1.1.cmml">=</mo><mn id="S4.T2.5.3.3.m1.1.1.3" xref="S4.T2.5.3.3.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.5.3.3.m1.1b"><apply id="S4.T2.5.3.3.m1.1.1.cmml" xref="S4.T2.5.3.3.m1.1.1"><eq id="S4.T2.5.3.3.m1.1.1.1.cmml" xref="S4.T2.5.3.3.m1.1.1.1"></eq><ci id="S4.T2.5.3.3.m1.1.1.2.cmml" xref="S4.T2.5.3.3.m1.1.1.2">𝑎</ci><cn id="S4.T2.5.3.3.m1.1.1.3.cmml" type="float" xref="S4.T2.5.3.3.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.5.3.3.m1.1c">a=0.005</annotation><annotation encoding="application/x-llamapun" id="S4.T2.5.3.3.m1.1d">italic_a = 0.005</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.5.4.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S4.T2.5.4.1.1">1</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.4.1.2">1.1799688</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.4.1.3">1.2293914</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.4.1.4">1.1847666</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.5.4.1.5">1.1823638</td> </tr> <tr class="ltx_tr" id="S4.T2.5.5.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S4.T2.5.5.2.1">2</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.5.2.2">1.6768750</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.5.2.3">1.7771792</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.5.2.4">1.6865131</td> <td class="ltx_td ltx_align_center" id="S4.T2.5.5.2.5">1.6816835</td> </tr> <tr class="ltx_tr" id="S4.T2.5.6.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S4.T2.5.6.3.1">3</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.6.3.2">2.0943951</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.6.3.3">2.0905692</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.6.3.4">2.0943568</td> <td class="ltx_td ltx_align_center" id="S4.T2.5.6.3.5">2.0943855</td> </tr> <tr class="ltx_tr" id="S4.T2.5.7.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S4.T2.5.7.4.1">4</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.7.4.2">2.7486684</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.7.4.3">2.8462967</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.7.4.4">2.7654332</td> <td class="ltx_td ltx_align_center" id="S4.T2.5.7.4.5">2.7570666</td> </tr> <tr class="ltx_tr" id="S4.T2.5.8.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S4.T2.5.8.5.1">5</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.8.5.2">2.8559933</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.8.5.3">2.9088003</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.5.8.5.4">2.8558962</td> <td class="ltx_td ltx_align_center" id="S4.T2.5.8.5.5">2.8559690</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S4.SS1.p12"> <p class="ltx_p" id="S4.SS1.p12.5">The vertex boundary conditions in Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.E31" title="In 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">31</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.E32" title="In 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">32</span></a>) lead to the following homogeneous system of linear equations:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx28"> <tbody id="S4.E35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle A_{i,j}f_{i,j}(N_{i,j})=\phi_{i},\quad B_{i,j}f_{i,j}(N_{i,j})=% \phi_{j}," class="ltx_Math" 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id="S4.E35.m1.12.12.2.3.cmml" xref="S4.E35.m1.12.12.2.4"><ci id="S4.E35.m1.11.11.1.1.cmml" xref="S4.E35.m1.11.11.1.1">𝑖</ci><ci id="S4.E35.m1.12.12.2.2.cmml" xref="S4.E35.m1.12.12.2.2">𝑗</ci></list></apply></apply><apply id="S4.E35.m1.13.13.1.1.2.2.3.cmml" xref="S4.E35.m1.13.13.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.E35.m1.13.13.1.1.2.2.3.1.cmml" xref="S4.E35.m1.13.13.1.1.2.2.3">subscript</csymbol><ci id="S4.E35.m1.13.13.1.1.2.2.3.2.cmml" xref="S4.E35.m1.13.13.1.1.2.2.3.2">italic-ϕ</ci><ci id="S4.E35.m1.13.13.1.1.2.2.3.3.cmml" xref="S4.E35.m1.13.13.1.1.2.2.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E35.m1.13c">\displaystyle A_{i,j}f_{i,j}(N_{i,j})=\phi_{i},\quad B_{i,j}f_{i,j}(N_{i,j})=% \phi_{j},</annotation><annotation encoding="application/x-llamapun" id="S4.E35.m1.13d">italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(35)</span></td> </tr></tbody> <tbody id="S4.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{j<i}C_{i,j}\bigg{\{}A_{i,j}f_{i,j}(1)+B_{i,j}\bigg{[}f_{i,j% }(N_{i,j}-1)-" class="ltx_math_unparsed" display="inline" id="S4.Ex26.m1.13"><semantics id="S4.Ex26.m1.13a"><mrow id="S4.Ex26.m1.13b"><mstyle displaystyle="true" id="S4.Ex26.m1.13.14"><munder id="S4.Ex26.m1.13.14a"><mo id="S4.Ex26.m1.13.14.2" movablelimits="false">∑</mo><mrow id="S4.Ex26.m1.13.14.3"><mi id="S4.Ex26.m1.13.14.3.2">j</mi><mo id="S4.Ex26.m1.13.14.3.1"><</mo><mi id="S4.Ex26.m1.13.14.3.3">i</mi></mrow></munder></mstyle><msub id="S4.Ex26.m1.13.15"><mi id="S4.Ex26.m1.13.15.2">C</mi><mrow id="S4.Ex26.m1.2.2.2.4"><mi id="S4.Ex26.m1.1.1.1.1">i</mi><mo id="S4.Ex26.m1.2.2.2.4.1">,</mo><mi id="S4.Ex26.m1.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex26.m1.13.16"><mo id="S4.Ex26.m1.13.16.1" maxsize="210%" minsize="210%">{</mo><msub id="S4.Ex26.m1.13.16.2"><mi id="S4.Ex26.m1.13.16.2.2">A</mi><mrow id="S4.Ex26.m1.4.4.2.4"><mi id="S4.Ex26.m1.3.3.1.1">i</mi><mo id="S4.Ex26.m1.4.4.2.4.1">,</mo><mi id="S4.Ex26.m1.4.4.2.2">j</mi></mrow></msub><msub id="S4.Ex26.m1.13.16.3"><mi id="S4.Ex26.m1.13.16.3.2">f</mi><mrow id="S4.Ex26.m1.6.6.2.4"><mi id="S4.Ex26.m1.5.5.1.1">i</mi><mo id="S4.Ex26.m1.6.6.2.4.1">,</mo><mi id="S4.Ex26.m1.6.6.2.2">j</mi></mrow></msub><mrow id="S4.Ex26.m1.13.16.4"><mo id="S4.Ex26.m1.13.16.4.1" stretchy="false">(</mo><mn id="S4.Ex26.m1.13.13">1</mn><mo id="S4.Ex26.m1.13.16.4.2" stretchy="false">)</mo></mrow><mo id="S4.Ex26.m1.13.16.5">+</mo><msub id="S4.Ex26.m1.13.16.6"><mi id="S4.Ex26.m1.13.16.6.2">B</mi><mrow id="S4.Ex26.m1.8.8.2.4"><mi id="S4.Ex26.m1.7.7.1.1">i</mi><mo id="S4.Ex26.m1.8.8.2.4.1">,</mo><mi id="S4.Ex26.m1.8.8.2.2">j</mi></mrow></msub><mrow id="S4.Ex26.m1.13.16.7"><mo id="S4.Ex26.m1.13.16.7.1" maxsize="210%" minsize="210%">[</mo><msub id="S4.Ex26.m1.13.16.7.2"><mi id="S4.Ex26.m1.13.16.7.2.2">f</mi><mrow id="S4.Ex26.m1.10.10.2.4"><mi id="S4.Ex26.m1.9.9.1.1">i</mi><mo id="S4.Ex26.m1.10.10.2.4.1">,</mo><mi id="S4.Ex26.m1.10.10.2.2">j</mi></mrow></msub><mrow id="S4.Ex26.m1.13.16.7.3"><mo id="S4.Ex26.m1.13.16.7.3.1" stretchy="false">(</mo><msub id="S4.Ex26.m1.13.16.7.3.2"><mi id="S4.Ex26.m1.13.16.7.3.2.2">N</mi><mrow id="S4.Ex26.m1.12.12.2.4"><mi id="S4.Ex26.m1.11.11.1.1">i</mi><mo id="S4.Ex26.m1.12.12.2.4.1">,</mo><mi id="S4.Ex26.m1.12.12.2.2">j</mi></mrow></msub><mo id="S4.Ex26.m1.13.16.7.3.3">−</mo><mn id="S4.Ex26.m1.13.16.7.3.4">1</mn><mo id="S4.Ex26.m1.13.16.7.3.5" stretchy="false">)</mo></mrow><mo id="S4.Ex26.m1.13.16.7.4">−</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex26.m1.13c">\displaystyle\sum_{j<i}C_{i,j}\bigg{\{}A_{i,j}f_{i,j}(1)+B_{i,j}\bigg{[}f_{i,j% }(N_{i,j}-1)-</annotation><annotation encoding="application/x-llamapun" id="S4.Ex26.m1.13d">∑ start_POSTSUBSCRIPT italic_j < italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( 1 ) + italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [ italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - 1 ) -</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle f_{i,j}(N_{i,j})\bigg{]}\bigg{\}}" class="ltx_math_unparsed" display="inline" id="S4.Ex26.m2.4"><semantics id="S4.Ex26.m2.4a"><mrow id="S4.Ex26.m2.4b"><mrow id="S4.Ex26.m2.4.5"><msub id="S4.Ex26.m2.4.5.1"><mi id="S4.Ex26.m2.4.5.1.2">f</mi><mrow id="S4.Ex26.m2.2.2.2.4"><mi id="S4.Ex26.m2.1.1.1.1">i</mi><mo id="S4.Ex26.m2.2.2.2.4.1">,</mo><mi id="S4.Ex26.m2.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex26.m2.4.5.2"><mo id="S4.Ex26.m2.4.5.2.1" stretchy="false">(</mo><msub id="S4.Ex26.m2.4.5.2.2"><mi id="S4.Ex26.m2.4.5.2.2.2">N</mi><mrow id="S4.Ex26.m2.4.4.2.4"><mi id="S4.Ex26.m2.3.3.1.1">i</mi><mo id="S4.Ex26.m2.4.4.2.4.1">,</mo><mi id="S4.Ex26.m2.4.4.2.2">j</mi></mrow></msub><mo id="S4.Ex26.m2.4.5.2.3" stretchy="false">)</mo></mrow><mo id="S4.Ex26.m2.4.5.3" maxsize="210%" minsize="210%">]</mo></mrow><mo id="S4.Ex26.m2.4.6" maxsize="210%" minsize="210%">}</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex26.m2.4c">\displaystyle f_{i,j}(N_{i,j})\bigg{]}\bigg{\}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex26.m2.4d">italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ] }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle+\sum_{j>i}C_{i,j}\bigg{\{}A_{i,j}\left[f_{i,j}(N_{i,j}-1)-f_{i,j% }(N_{i,j})\right]+" class="ltx_math_unparsed" display="inline" id="S4.Ex27.m1.12"><semantics id="S4.Ex27.m1.12a"><mrow id="S4.Ex27.m1.12b"><mo id="S4.Ex27.m1.12.13">+</mo><mstyle displaystyle="true" id="S4.Ex27.m1.12.14"><munder id="S4.Ex27.m1.12.14a"><mo id="S4.Ex27.m1.12.14.2" movablelimits="false">∑</mo><mrow id="S4.Ex27.m1.12.14.3"><mi id="S4.Ex27.m1.12.14.3.2">j</mi><mo id="S4.Ex27.m1.12.14.3.1">></mo><mi id="S4.Ex27.m1.12.14.3.3">i</mi></mrow></munder></mstyle><msub id="S4.Ex27.m1.12.15"><mi id="S4.Ex27.m1.12.15.2">C</mi><mrow id="S4.Ex27.m1.2.2.2.4"><mi id="S4.Ex27.m1.1.1.1.1">i</mi><mo id="S4.Ex27.m1.2.2.2.4.1">,</mo><mi id="S4.Ex27.m1.2.2.2.2">j</mi></mrow></msub><mrow id="S4.Ex27.m1.12.16"><mo id="S4.Ex27.m1.12.16.1" maxsize="210%" minsize="210%">{</mo><msub id="S4.Ex27.m1.12.16.2"><mi id="S4.Ex27.m1.12.16.2.2">A</mi><mrow id="S4.Ex27.m1.4.4.2.4"><mi id="S4.Ex27.m1.3.3.1.1">i</mi><mo id="S4.Ex27.m1.4.4.2.4.1">,</mo><mi id="S4.Ex27.m1.4.4.2.2">j</mi></mrow></msub><mrow id="S4.Ex27.m1.12.16.3"><mo id="S4.Ex27.m1.12.16.3.1">[</mo><msub id="S4.Ex27.m1.12.16.3.2"><mi id="S4.Ex27.m1.12.16.3.2.2">f</mi><mrow id="S4.Ex27.m1.6.6.2.4"><mi id="S4.Ex27.m1.5.5.1.1">i</mi><mo id="S4.Ex27.m1.6.6.2.4.1">,</mo><mi id="S4.Ex27.m1.6.6.2.2">j</mi></mrow></msub><mrow id="S4.Ex27.m1.12.16.3.3"><mo id="S4.Ex27.m1.12.16.3.3.1" stretchy="false">(</mo><msub id="S4.Ex27.m1.12.16.3.3.2"><mi id="S4.Ex27.m1.12.16.3.3.2.2">N</mi><mrow id="S4.Ex27.m1.8.8.2.4"><mi id="S4.Ex27.m1.7.7.1.1">i</mi><mo id="S4.Ex27.m1.8.8.2.4.1">,</mo><mi id="S4.Ex27.m1.8.8.2.2">j</mi></mrow></msub><mo id="S4.Ex27.m1.12.16.3.3.3">−</mo><mn id="S4.Ex27.m1.12.16.3.3.4">1</mn><mo id="S4.Ex27.m1.12.16.3.3.5" stretchy="false">)</mo></mrow><mo id="S4.Ex27.m1.12.16.3.4">−</mo><msub id="S4.Ex27.m1.12.16.3.5"><mi id="S4.Ex27.m1.12.16.3.5.2">f</mi><mrow id="S4.Ex27.m1.10.10.2.4"><mi id="S4.Ex27.m1.9.9.1.1">i</mi><mo id="S4.Ex27.m1.10.10.2.4.1">,</mo><mi id="S4.Ex27.m1.10.10.2.2">j</mi></mrow></msub><mrow id="S4.Ex27.m1.12.16.3.6"><mo id="S4.Ex27.m1.12.16.3.6.1" stretchy="false">(</mo><msub id="S4.Ex27.m1.12.16.3.6.2"><mi id="S4.Ex27.m1.12.16.3.6.2.2">N</mi><mrow id="S4.Ex27.m1.12.12.2.4"><mi id="S4.Ex27.m1.11.11.1.1">i</mi><mo id="S4.Ex27.m1.12.12.2.4.1">,</mo><mi id="S4.Ex27.m1.12.12.2.2">j</mi></mrow></msub><mo id="S4.Ex27.m1.12.16.3.6.3" stretchy="false">)</mo></mrow><mo id="S4.Ex27.m1.12.16.3.7">]</mo></mrow><mo id="S4.Ex27.m1.12.16.4">+</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex27.m1.12c">\displaystyle+\sum_{j>i}C_{i,j}\bigg{\{}A_{i,j}\left[f_{i,j}(N_{i,j}-1)-f_{i,j% }(N_{i,j})\right]+</annotation><annotation encoding="application/x-llamapun" id="S4.Ex27.m1.12d">+ ∑ start_POSTSUBSCRIPT italic_j > italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT [ italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT - 1 ) - italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) ] +</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle B_{i,j}f_{i,j}(1)\bigg{\}}" class="ltx_math_unparsed" display="inline" id="S4.Ex27.m2.5"><semantics id="S4.Ex27.m2.5a"><mrow id="S4.Ex27.m2.5b"><msub id="S4.Ex27.m2.5.6"><mi id="S4.Ex27.m2.5.6.2">B</mi><mrow id="S4.Ex27.m2.2.2.2.4"><mi id="S4.Ex27.m2.1.1.1.1">i</mi><mo id="S4.Ex27.m2.2.2.2.4.1">,</mo><mi id="S4.Ex27.m2.2.2.2.2">j</mi></mrow></msub><msub id="S4.Ex27.m2.5.7"><mi id="S4.Ex27.m2.5.7.2">f</mi><mrow id="S4.Ex27.m2.4.4.2.4"><mi id="S4.Ex27.m2.3.3.1.1">i</mi><mo id="S4.Ex27.m2.4.4.2.4.1">,</mo><mi id="S4.Ex27.m2.4.4.2.2">j</mi></mrow></msub><mrow id="S4.Ex27.m2.5.8"><mo id="S4.Ex27.m2.5.8.1" stretchy="false">(</mo><mn id="S4.Ex27.m2.5.5">1</mn><mo id="S4.Ex27.m2.5.8.2" stretchy="false">)</mo></mrow><mo id="S4.Ex27.m2.5.9" maxsize="210%" minsize="210%">}</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex27.m2.5c">\displaystyle B_{i,j}f_{i,j}(1)\bigg{\}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex27.m2.5d">italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( 1 ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S4.E36.m1.1"><semantics id="S4.E36.m1.1a"><mo id="S4.E36.m1.1.1" xref="S4.E36.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.E36.m1.1b"><eq id="S4.E36.m1.1.1.cmml" xref="S4.E36.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.E36.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S4.E36.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lambda_{i}\phi_{i}." class="ltx_Math" display="inline" id="S4.E36.m2.1"><semantics id="S4.E36.m2.1a"><mrow id="S4.E36.m2.1.1.1" xref="S4.E36.m2.1.1.1.1.cmml"><mrow id="S4.E36.m2.1.1.1.1" xref="S4.E36.m2.1.1.1.1.cmml"><msub id="S4.E36.m2.1.1.1.1.2" xref="S4.E36.m2.1.1.1.1.2.cmml"><mi id="S4.E36.m2.1.1.1.1.2.2" xref="S4.E36.m2.1.1.1.1.2.2.cmml">λ</mi><mi id="S4.E36.m2.1.1.1.1.2.3" xref="S4.E36.m2.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S4.E36.m2.1.1.1.1.1" xref="S4.E36.m2.1.1.1.1.1.cmml">⁢</mo><msub id="S4.E36.m2.1.1.1.1.3" xref="S4.E36.m2.1.1.1.1.3.cmml"><mi id="S4.E36.m2.1.1.1.1.3.2" xref="S4.E36.m2.1.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.E36.m2.1.1.1.1.3.3" xref="S4.E36.m2.1.1.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S4.E36.m2.1.1.1.2" lspace="0em" xref="S4.E36.m2.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E36.m2.1b"><apply id="S4.E36.m2.1.1.1.1.cmml" xref="S4.E36.m2.1.1.1"><times id="S4.E36.m2.1.1.1.1.1.cmml" xref="S4.E36.m2.1.1.1.1.1"></times><apply id="S4.E36.m2.1.1.1.1.2.cmml" xref="S4.E36.m2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E36.m2.1.1.1.1.2.1.cmml" xref="S4.E36.m2.1.1.1.1.2">subscript</csymbol><ci id="S4.E36.m2.1.1.1.1.2.2.cmml" xref="S4.E36.m2.1.1.1.1.2.2">𝜆</ci><ci id="S4.E36.m2.1.1.1.1.2.3.cmml" xref="S4.E36.m2.1.1.1.1.2.3">𝑖</ci></apply><apply id="S4.E36.m2.1.1.1.1.3.cmml" xref="S4.E36.m2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E36.m2.1.1.1.1.3.1.cmml" xref="S4.E36.m2.1.1.1.1.3">subscript</csymbol><ci id="S4.E36.m2.1.1.1.1.3.2.cmml" xref="S4.E36.m2.1.1.1.1.3.2">italic-ϕ</ci><ci id="S4.E36.m2.1.1.1.1.3.3.cmml" xref="S4.E36.m2.1.1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E36.m2.1c">\displaystyle\lambda_{i}\phi_{i}.</annotation><annotation encoding="application/x-llamapun" id="S4.E36.m2.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(36)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p12.3">where <math alttext="f_{i,j}(n_{i,j})=g_{+}(a_{i,j})^{n_{i,j}}-g_{-}(a_{i,j})^{n_{i,j}}" class="ltx_Math" display="inline" id="S4.SS1.p12.1.m1.15"><semantics id="S4.SS1.p12.1.m1.15a"><mrow id="S4.SS1.p12.1.m1.15.15" xref="S4.SS1.p12.1.m1.15.15.cmml"><mrow id="S4.SS1.p12.1.m1.13.13.1" xref="S4.SS1.p12.1.m1.13.13.1.cmml"><msub id="S4.SS1.p12.1.m1.13.13.1.3" xref="S4.SS1.p12.1.m1.13.13.1.3.cmml"><mi id="S4.SS1.p12.1.m1.13.13.1.3.2" xref="S4.SS1.p12.1.m1.13.13.1.3.2.cmml">f</mi><mrow id="S4.SS1.p12.1.m1.2.2.2.4" xref="S4.SS1.p12.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p12.1.m1.1.1.1.1" xref="S4.SS1.p12.1.m1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p12.1.m1.2.2.2.4.1" xref="S4.SS1.p12.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p12.1.m1.2.2.2.2" xref="S4.SS1.p12.1.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p12.1.m1.13.13.1.2" xref="S4.SS1.p12.1.m1.13.13.1.2.cmml">⁢</mo><mrow id="S4.SS1.p12.1.m1.13.13.1.1.1" xref="S4.SS1.p12.1.m1.13.13.1.1.1.1.cmml"><mo id="S4.SS1.p12.1.m1.13.13.1.1.1.2" stretchy="false" xref="S4.SS1.p12.1.m1.13.13.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p12.1.m1.13.13.1.1.1.1" xref="S4.SS1.p12.1.m1.13.13.1.1.1.1.cmml"><mi id="S4.SS1.p12.1.m1.13.13.1.1.1.1.2" xref="S4.SS1.p12.1.m1.13.13.1.1.1.1.2.cmml">n</mi><mrow id="S4.SS1.p12.1.m1.4.4.2.4" xref="S4.SS1.p12.1.m1.4.4.2.3.cmml"><mi id="S4.SS1.p12.1.m1.3.3.1.1" xref="S4.SS1.p12.1.m1.3.3.1.1.cmml">i</mi><mo id="S4.SS1.p12.1.m1.4.4.2.4.1" xref="S4.SS1.p12.1.m1.4.4.2.3.cmml">,</mo><mi id="S4.SS1.p12.1.m1.4.4.2.2" xref="S4.SS1.p12.1.m1.4.4.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p12.1.m1.13.13.1.1.1.3" stretchy="false" xref="S4.SS1.p12.1.m1.13.13.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p12.1.m1.15.15.4" xref="S4.SS1.p12.1.m1.15.15.4.cmml">=</mo><mrow id="S4.SS1.p12.1.m1.15.15.3" xref="S4.SS1.p12.1.m1.15.15.3.cmml"><mrow id="S4.SS1.p12.1.m1.14.14.2.1" xref="S4.SS1.p12.1.m1.14.14.2.1.cmml"><msub id="S4.SS1.p12.1.m1.14.14.2.1.3" xref="S4.SS1.p12.1.m1.14.14.2.1.3.cmml"><mi id="S4.SS1.p12.1.m1.14.14.2.1.3.2" xref="S4.SS1.p12.1.m1.14.14.2.1.3.2.cmml">g</mi><mo id="S4.SS1.p12.1.m1.14.14.2.1.3.3" xref="S4.SS1.p12.1.m1.14.14.2.1.3.3.cmml">+</mo></msub><mo id="S4.SS1.p12.1.m1.14.14.2.1.2" xref="S4.SS1.p12.1.m1.14.14.2.1.2.cmml">⁢</mo><msup id="S4.SS1.p12.1.m1.14.14.2.1.1" xref="S4.SS1.p12.1.m1.14.14.2.1.1.cmml"><mrow id="S4.SS1.p12.1.m1.14.14.2.1.1.1.1" xref="S4.SS1.p12.1.m1.14.14.2.1.1.1.1.1.cmml"><mo id="S4.SS1.p12.1.m1.14.14.2.1.1.1.1.2" stretchy="false" xref="S4.SS1.p12.1.m1.14.14.2.1.1.1.1.1.cmml">(</mo><msub 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xref="S4.SS1.p12.1.m1.8.8.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p12.1.m1.8.8.2.2.2.2" xref="S4.SS1.p12.1.m1.8.8.2.2.2.2.cmml">j</mi></mrow></msub></msup></mrow><mo id="S4.SS1.p12.1.m1.15.15.3.3" xref="S4.SS1.p12.1.m1.15.15.3.3.cmml">−</mo><mrow id="S4.SS1.p12.1.m1.15.15.3.2" xref="S4.SS1.p12.1.m1.15.15.3.2.cmml"><msub id="S4.SS1.p12.1.m1.15.15.3.2.3" xref="S4.SS1.p12.1.m1.15.15.3.2.3.cmml"><mi id="S4.SS1.p12.1.m1.15.15.3.2.3.2" xref="S4.SS1.p12.1.m1.15.15.3.2.3.2.cmml">g</mi><mo id="S4.SS1.p12.1.m1.15.15.3.2.3.3" xref="S4.SS1.p12.1.m1.15.15.3.2.3.3.cmml">−</mo></msub><mo id="S4.SS1.p12.1.m1.15.15.3.2.2" xref="S4.SS1.p12.1.m1.15.15.3.2.2.cmml">⁢</mo><msup id="S4.SS1.p12.1.m1.15.15.3.2.1" xref="S4.SS1.p12.1.m1.15.15.3.2.1.cmml"><mrow id="S4.SS1.p12.1.m1.15.15.3.2.1.1.1" xref="S4.SS1.p12.1.m1.15.15.3.2.1.1.1.1.cmml"><mo id="S4.SS1.p12.1.m1.15.15.3.2.1.1.1.2" stretchy="false" xref="S4.SS1.p12.1.m1.15.15.3.2.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p12.1.m1.15.15.3.2.1.1.1.1" 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start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. This is a system of homogeneous equations for <math alttext="A_{i,j},\;B_{i,j}" class="ltx_Math" display="inline" id="S4.SS1.p12.2.m2.6"><semantics id="S4.SS1.p12.2.m2.6a"><mrow id="S4.SS1.p12.2.m2.6.6.2" xref="S4.SS1.p12.2.m2.6.6.3.cmml"><msub id="S4.SS1.p12.2.m2.5.5.1.1" xref="S4.SS1.p12.2.m2.5.5.1.1.cmml"><mi id="S4.SS1.p12.2.m2.5.5.1.1.2" xref="S4.SS1.p12.2.m2.5.5.1.1.2.cmml">A</mi><mrow id="S4.SS1.p12.2.m2.2.2.2.4" xref="S4.SS1.p12.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p12.2.m2.1.1.1.1" xref="S4.SS1.p12.2.m2.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p12.2.m2.2.2.2.4.1" xref="S4.SS1.p12.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p12.2.m2.2.2.2.2" xref="S4.SS1.p12.2.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS1.p12.2.m2.6.6.2.3" rspace="0.447em" xref="S4.SS1.p12.2.m2.6.6.3.cmml">,</mo><msub id="S4.SS1.p12.2.m2.6.6.2.2" xref="S4.SS1.p12.2.m2.6.6.2.2.cmml"><mi id="S4.SS1.p12.2.m2.6.6.2.2.2" xref="S4.SS1.p12.2.m2.6.6.2.2.2.cmml">B</mi><mrow id="S4.SS1.p12.2.m2.4.4.2.4" 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cd="ambiguous" id="S4.SS1.p12.2.m2.6.6.2.2.1.cmml" xref="S4.SS1.p12.2.m2.6.6.2.2">subscript</csymbol><ci id="S4.SS1.p12.2.m2.6.6.2.2.2.cmml" xref="S4.SS1.p12.2.m2.6.6.2.2.2">𝐵</ci><list id="S4.SS1.p12.2.m2.4.4.2.3.cmml" xref="S4.SS1.p12.2.m2.4.4.2.4"><ci id="S4.SS1.p12.2.m2.3.3.1.1.cmml" xref="S4.SS1.p12.2.m2.3.3.1.1">𝑖</ci><ci id="S4.SS1.p12.2.m2.4.4.2.2.cmml" xref="S4.SS1.p12.2.m2.4.4.2.2">𝑗</ci></list></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p12.2.m2.6c">A_{i,j},\;B_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p12.2.m2.6d">italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\phi_{i}" class="ltx_Math" display="inline" id="S4.SS1.p12.3.m3.1"><semantics id="S4.SS1.p12.3.m3.1a"><msub id="S4.SS1.p12.3.m3.1.1" xref="S4.SS1.p12.3.m3.1.1.cmml"><mi id="S4.SS1.p12.3.m3.1.1.2" xref="S4.SS1.p12.3.m3.1.1.2.cmml">ϕ</mi><mi id="S4.SS1.p12.3.m3.1.1.3" xref="S4.SS1.p12.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p12.3.m3.1b"><apply id="S4.SS1.p12.3.m3.1.1.cmml" xref="S4.SS1.p12.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p12.3.m3.1.1.1.cmml" xref="S4.SS1.p12.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.p12.3.m3.1.1.2.cmml" xref="S4.SS1.p12.3.m3.1.1.2">italic-ϕ</ci><ci id="S4.SS1.p12.3.m3.1.1.3.cmml" xref="S4.SS1.p12.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p12.3.m3.1c">\phi_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p12.3.m3.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which has a non-trivial solution when</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx29"> <tbody id="S4.E37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\det(M(k))=0," class="ltx_Math" display="inline" id="S4.E37.m1.2"><semantics id="S4.E37.m1.2a"><mrow id="S4.E37.m1.2.2.1" xref="S4.E37.m1.2.2.1.1.cmml"><mrow id="S4.E37.m1.2.2.1.1" xref="S4.E37.m1.2.2.1.1.cmml"><mrow id="S4.E37.m1.2.2.1.1.1" xref="S4.E37.m1.2.2.1.1.1.cmml"><mo id="S4.E37.m1.2.2.1.1.1.2" movablelimits="false" rspace="0em" xref="S4.E37.m1.2.2.1.1.1.2.cmml">det</mo><mrow id="S4.E37.m1.2.2.1.1.1.1.1" xref="S4.E37.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S4.E37.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.E37.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E37.m1.2.2.1.1.1.1.1.1" xref="S4.E37.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S4.E37.m1.2.2.1.1.1.1.1.1.2" xref="S4.E37.m1.2.2.1.1.1.1.1.1.2.cmml">M</mi><mo id="S4.E37.m1.2.2.1.1.1.1.1.1.1" xref="S4.E37.m1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E37.m1.2.2.1.1.1.1.1.1.3.2" xref="S4.E37.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S4.E37.m1.2.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.E37.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mi id="S4.E37.m1.1.1" xref="S4.E37.m1.1.1.cmml">k</mi><mo id="S4.E37.m1.2.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.E37.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E37.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.E37.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E37.m1.2.2.1.1.2" xref="S4.E37.m1.2.2.1.1.2.cmml">=</mo><mn id="S4.E37.m1.2.2.1.1.3" xref="S4.E37.m1.2.2.1.1.3.cmml">0</mn></mrow><mo id="S4.E37.m1.2.2.1.2" xref="S4.E37.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E37.m1.2b"><apply id="S4.E37.m1.2.2.1.1.cmml" xref="S4.E37.m1.2.2.1"><eq id="S4.E37.m1.2.2.1.1.2.cmml" xref="S4.E37.m1.2.2.1.1.2"></eq><apply id="S4.E37.m1.2.2.1.1.1.cmml" xref="S4.E37.m1.2.2.1.1.1"><determinant id="S4.E37.m1.2.2.1.1.1.2.cmml" xref="S4.E37.m1.2.2.1.1.1.2"></determinant><apply id="S4.E37.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E37.m1.2.2.1.1.1.1.1"><times id="S4.E37.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E37.m1.2.2.1.1.1.1.1.1.1"></times><ci id="S4.E37.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E37.m1.2.2.1.1.1.1.1.1.2">𝑀</ci><ci id="S4.E37.m1.1.1.cmml" xref="S4.E37.m1.1.1">𝑘</ci></apply></apply><cn id="S4.E37.m1.2.2.1.1.3.cmml" type="integer" xref="S4.E37.m1.2.2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E37.m1.2c">\displaystyle\det(M(k))=0,</annotation><annotation encoding="application/x-llamapun" id="S4.E37.m1.2d">roman_det ( italic_M ( italic_k ) ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(37)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p12.4">where <math alttext="M(k)" class="ltx_Math" display="inline" id="S4.SS1.p12.4.m1.1"><semantics id="S4.SS1.p12.4.m1.1a"><mrow id="S4.SS1.p12.4.m1.1.2" xref="S4.SS1.p12.4.m1.1.2.cmml"><mi id="S4.SS1.p12.4.m1.1.2.2" xref="S4.SS1.p12.4.m1.1.2.2.cmml">M</mi><mo id="S4.SS1.p12.4.m1.1.2.1" xref="S4.SS1.p12.4.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p12.4.m1.1.2.3.2" xref="S4.SS1.p12.4.m1.1.2.cmml"><mo id="S4.SS1.p12.4.m1.1.2.3.2.1" stretchy="false" xref="S4.SS1.p12.4.m1.1.2.cmml">(</mo><mi id="S4.SS1.p12.4.m1.1.1" xref="S4.SS1.p12.4.m1.1.1.cmml">k</mi><mo id="S4.SS1.p12.4.m1.1.2.3.2.2" stretchy="false" xref="S4.SS1.p12.4.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p12.4.m1.1b"><apply id="S4.SS1.p12.4.m1.1.2.cmml" xref="S4.SS1.p12.4.m1.1.2"><times id="S4.SS1.p12.4.m1.1.2.1.cmml" xref="S4.SS1.p12.4.m1.1.2.1"></times><ci id="S4.SS1.p12.4.m1.1.2.2.cmml" xref="S4.SS1.p12.4.m1.1.2.2">𝑀</ci><ci id="S4.SS1.p12.4.m1.1.1.cmml" xref="S4.SS1.p12.4.m1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p12.4.m1.1c">M(k)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p12.4.m1.1d">italic_M ( italic_k )</annotation></semantics></math> matrix has the same size and structure as in the section <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3" title="3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Star branched lattice</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.3">In this subsection, we consider quantum star graph with three discrete edges (see, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.F2" title="Figure 2 ‣ 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a>). For such a graph, <math alttext="N_{1,j}a_{1,j}=L_{1,j}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.6"><semantics id="S4.SS2.p1.1.m1.6a"><mrow id="S4.SS2.p1.1.m1.6.7" xref="S4.SS2.p1.1.m1.6.7.cmml"><mrow id="S4.SS2.p1.1.m1.6.7.2" xref="S4.SS2.p1.1.m1.6.7.2.cmml"><msub id="S4.SS2.p1.1.m1.6.7.2.2" xref="S4.SS2.p1.1.m1.6.7.2.2.cmml"><mi id="S4.SS2.p1.1.m1.6.7.2.2.2" xref="S4.SS2.p1.1.m1.6.7.2.2.2.cmml">N</mi><mrow id="S4.SS2.p1.1.m1.2.2.2.4" xref="S4.SS2.p1.1.m1.2.2.2.3.cmml"><mn id="S4.SS2.p1.1.m1.1.1.1.1" xref="S4.SS2.p1.1.m1.1.1.1.1.cmml">1</mn><mo id="S4.SS2.p1.1.m1.2.2.2.4.1" xref="S4.SS2.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p1.1.m1.2.2.2.2" xref="S4.SS2.p1.1.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS2.p1.1.m1.6.7.2.1" xref="S4.SS2.p1.1.m1.6.7.2.1.cmml">⁢</mo><msub id="S4.SS2.p1.1.m1.6.7.2.3" xref="S4.SS2.p1.1.m1.6.7.2.3.cmml"><mi id="S4.SS2.p1.1.m1.6.7.2.3.2" xref="S4.SS2.p1.1.m1.6.7.2.3.2.cmml">a</mi><mrow id="S4.SS2.p1.1.m1.4.4.2.4" xref="S4.SS2.p1.1.m1.4.4.2.3.cmml"><mn id="S4.SS2.p1.1.m1.3.3.1.1" xref="S4.SS2.p1.1.m1.3.3.1.1.cmml">1</mn><mo id="S4.SS2.p1.1.m1.4.4.2.4.1" xref="S4.SS2.p1.1.m1.4.4.2.3.cmml">,</mo><mi id="S4.SS2.p1.1.m1.4.4.2.2" xref="S4.SS2.p1.1.m1.4.4.2.2.cmml">j</mi></mrow></msub></mrow><mo id="S4.SS2.p1.1.m1.6.7.1" xref="S4.SS2.p1.1.m1.6.7.1.cmml">=</mo><msub id="S4.SS2.p1.1.m1.6.7.3" xref="S4.SS2.p1.1.m1.6.7.3.cmml"><mi id="S4.SS2.p1.1.m1.6.7.3.2" xref="S4.SS2.p1.1.m1.6.7.3.2.cmml">L</mi><mrow id="S4.SS2.p1.1.m1.6.6.2.4" xref="S4.SS2.p1.1.m1.6.6.2.3.cmml"><mn id="S4.SS2.p1.1.m1.5.5.1.1" xref="S4.SS2.p1.1.m1.5.5.1.1.cmml">1</mn><mo id="S4.SS2.p1.1.m1.6.6.2.4.1" xref="S4.SS2.p1.1.m1.6.6.2.3.cmml">,</mo><mi id="S4.SS2.p1.1.m1.6.6.2.2" xref="S4.SS2.p1.1.m1.6.6.2.2.cmml">j</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.6b"><apply id="S4.SS2.p1.1.m1.6.7.cmml" xref="S4.SS2.p1.1.m1.6.7"><eq id="S4.SS2.p1.1.m1.6.7.1.cmml" xref="S4.SS2.p1.1.m1.6.7.1"></eq><apply id="S4.SS2.p1.1.m1.6.7.2.cmml" xref="S4.SS2.p1.1.m1.6.7.2"><times id="S4.SS2.p1.1.m1.6.7.2.1.cmml" xref="S4.SS2.p1.1.m1.6.7.2.1"></times><apply id="S4.SS2.p1.1.m1.6.7.2.2.cmml" xref="S4.SS2.p1.1.m1.6.7.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.6.7.2.2.1.cmml" xref="S4.SS2.p1.1.m1.6.7.2.2">subscript</csymbol><ci id="S4.SS2.p1.1.m1.6.7.2.2.2.cmml" xref="S4.SS2.p1.1.m1.6.7.2.2.2">𝑁</ci><list id="S4.SS2.p1.1.m1.2.2.2.3.cmml" xref="S4.SS2.p1.1.m1.2.2.2.4"><cn id="S4.SS2.p1.1.m1.1.1.1.1.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.1.1">1</cn><ci id="S4.SS2.p1.1.m1.2.2.2.2.cmml" xref="S4.SS2.p1.1.m1.2.2.2.2">𝑗</ci></list></apply><apply id="S4.SS2.p1.1.m1.6.7.2.3.cmml" xref="S4.SS2.p1.1.m1.6.7.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.6.7.2.3.1.cmml" xref="S4.SS2.p1.1.m1.6.7.2.3">subscript</csymbol><ci id="S4.SS2.p1.1.m1.6.7.2.3.2.cmml" xref="S4.SS2.p1.1.m1.6.7.2.3.2">𝑎</ci><list id="S4.SS2.p1.1.m1.4.4.2.3.cmml" xref="S4.SS2.p1.1.m1.4.4.2.4"><cn id="S4.SS2.p1.1.m1.3.3.1.1.cmml" type="integer" xref="S4.SS2.p1.1.m1.3.3.1.1">1</cn><ci id="S4.SS2.p1.1.m1.4.4.2.2.cmml" xref="S4.SS2.p1.1.m1.4.4.2.2">𝑗</ci></list></apply></apply><apply id="S4.SS2.p1.1.m1.6.7.3.cmml" xref="S4.SS2.p1.1.m1.6.7.3"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.6.7.3.1.cmml" xref="S4.SS2.p1.1.m1.6.7.3">subscript</csymbol><ci id="S4.SS2.p1.1.m1.6.7.3.2.cmml" xref="S4.SS2.p1.1.m1.6.7.3.2">𝐿</ci><list id="S4.SS2.p1.1.m1.6.6.2.3.cmml" xref="S4.SS2.p1.1.m1.6.6.2.4"><cn id="S4.SS2.p1.1.m1.5.5.1.1.cmml" type="integer" xref="S4.SS2.p1.1.m1.5.5.1.1">1</cn><ci id="S4.SS2.p1.1.m1.6.6.2.2.cmml" xref="S4.SS2.p1.1.m1.6.6.2.2">𝑗</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.6c">N_{1,j}a_{1,j}=L_{1,j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.6d">italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is the length of the edge the <math alttext="(1,j)" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.2"><semantics id="S4.SS2.p1.2.m2.2a"><mrow id="S4.SS2.p1.2.m2.2.3.2" xref="S4.SS2.p1.2.m2.2.3.1.cmml"><mo id="S4.SS2.p1.2.m2.2.3.2.1" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.1.cmml">(</mo><mn id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml">1</mn><mo id="S4.SS2.p1.2.m2.2.3.2.2" xref="S4.SS2.p1.2.m2.2.3.1.cmml">,</mo><mi id="S4.SS2.p1.2.m2.2.2" xref="S4.SS2.p1.2.m2.2.2.cmml">j</mi><mo id="S4.SS2.p1.2.m2.2.3.2.3" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.2b"><interval closure="open" id="S4.SS2.p1.2.m2.2.3.1.cmml" xref="S4.SS2.p1.2.m2.2.3.2"><cn id="S4.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1">1</cn><ci id="S4.SS2.p1.2.m2.2.2.cmml" xref="S4.SS2.p1.2.m2.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.2c">(1,j)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.2d">( 1 , italic_j )</annotation></semantics></math> for <math alttext="j=2,3,4" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.3"><semantics id="S4.SS2.p1.3.m3.3a"><mrow id="S4.SS2.p1.3.m3.3.4" xref="S4.SS2.p1.3.m3.3.4.cmml"><mi id="S4.SS2.p1.3.m3.3.4.2" xref="S4.SS2.p1.3.m3.3.4.2.cmml">j</mi><mo id="S4.SS2.p1.3.m3.3.4.1" xref="S4.SS2.p1.3.m3.3.4.1.cmml">=</mo><mrow id="S4.SS2.p1.3.m3.3.4.3.2" xref="S4.SS2.p1.3.m3.3.4.3.1.cmml"><mn id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml">2</mn><mo id="S4.SS2.p1.3.m3.3.4.3.2.1" xref="S4.SS2.p1.3.m3.3.4.3.1.cmml">,</mo><mn id="S4.SS2.p1.3.m3.2.2" xref="S4.SS2.p1.3.m3.2.2.cmml">3</mn><mo id="S4.SS2.p1.3.m3.3.4.3.2.2" xref="S4.SS2.p1.3.m3.3.4.3.1.cmml">,</mo><mn id="S4.SS2.p1.3.m3.3.3" xref="S4.SS2.p1.3.m3.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.3b"><apply id="S4.SS2.p1.3.m3.3.4.cmml" xref="S4.SS2.p1.3.m3.3.4"><eq id="S4.SS2.p1.3.m3.3.4.1.cmml" xref="S4.SS2.p1.3.m3.3.4.1"></eq><ci id="S4.SS2.p1.3.m3.3.4.2.cmml" xref="S4.SS2.p1.3.m3.3.4.2">𝑗</ci><list id="S4.SS2.p1.3.m3.3.4.3.1.cmml" xref="S4.SS2.p1.3.m3.3.4.3.2"><cn id="S4.SS2.p1.3.m3.1.1.cmml" type="integer" xref="S4.SS2.p1.3.m3.1.1">2</cn><cn id="S4.SS2.p1.3.m3.2.2.cmml" type="integer" xref="S4.SS2.p1.3.m3.2.2">3</cn><cn id="S4.SS2.p1.3.m3.3.3.cmml" type="integer" xref="S4.SS2.p1.3.m3.3.3">4</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.3c">j=2,3,4</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.3d">italic_j = 2 , 3 , 4</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.2">We choose <math alttext="\lambda_{1}=\lambda_{2}=\lambda_{3}=\lambda_{4}=0." class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mrow id="S4.SS2.p2.1.m1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.1.cmml"><mrow id="S4.SS2.p2.1.m1.1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.1.cmml"><msub id="S4.SS2.p2.1.m1.1.1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.1.1.2.cmml"><mi id="S4.SS2.p2.1.m1.1.1.1.1.2.2" xref="S4.SS2.p2.1.m1.1.1.1.1.2.2.cmml">λ</mi><mn id="S4.SS2.p2.1.m1.1.1.1.1.2.3" xref="S4.SS2.p2.1.m1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.p2.1.m1.1.1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.1.1.3.cmml">=</mo><msub id="S4.SS2.p2.1.m1.1.1.1.1.4" xref="S4.SS2.p2.1.m1.1.1.1.1.4.cmml"><mi id="S4.SS2.p2.1.m1.1.1.1.1.4.2" xref="S4.SS2.p2.1.m1.1.1.1.1.4.2.cmml">λ</mi><mn id="S4.SS2.p2.1.m1.1.1.1.1.4.3" xref="S4.SS2.p2.1.m1.1.1.1.1.4.3.cmml">2</mn></msub><mo id="S4.SS2.p2.1.m1.1.1.1.1.5" xref="S4.SS2.p2.1.m1.1.1.1.1.5.cmml">=</mo><msub id="S4.SS2.p2.1.m1.1.1.1.1.6" xref="S4.SS2.p2.1.m1.1.1.1.1.6.cmml"><mi id="S4.SS2.p2.1.m1.1.1.1.1.6.2" xref="S4.SS2.p2.1.m1.1.1.1.1.6.2.cmml">λ</mi><mn id="S4.SS2.p2.1.m1.1.1.1.1.6.3" xref="S4.SS2.p2.1.m1.1.1.1.1.6.3.cmml">3</mn></msub><mo id="S4.SS2.p2.1.m1.1.1.1.1.7" xref="S4.SS2.p2.1.m1.1.1.1.1.7.cmml">=</mo><msub id="S4.SS2.p2.1.m1.1.1.1.1.8" xref="S4.SS2.p2.1.m1.1.1.1.1.8.cmml"><mi id="S4.SS2.p2.1.m1.1.1.1.1.8.2" xref="S4.SS2.p2.1.m1.1.1.1.1.8.2.cmml">λ</mi><mn id="S4.SS2.p2.1.m1.1.1.1.1.8.3" xref="S4.SS2.p2.1.m1.1.1.1.1.8.3.cmml">4</mn></msub><mo id="S4.SS2.p2.1.m1.1.1.1.1.9" xref="S4.SS2.p2.1.m1.1.1.1.1.9.cmml">=</mo><mn id="S4.SS2.p2.1.m1.1.1.1.1.10" xref="S4.SS2.p2.1.m1.1.1.1.1.10.cmml">0</mn></mrow><mo id="S4.SS2.p2.1.m1.1.1.1.2" lspace="0em" xref="S4.SS2.p2.1.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1"><and id="S4.SS2.p2.1.m1.1.1.1.1a.cmml" xref="S4.SS2.p2.1.m1.1.1.1"></and><apply id="S4.SS2.p2.1.m1.1.1.1.1b.cmml" xref="S4.SS2.p2.1.m1.1.1.1"><eq id="S4.SS2.p2.1.m1.1.1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.3"></eq><apply id="S4.SS2.p2.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.1.1.2.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.1.1.2.2.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.2.2">𝜆</ci><cn id="S4.SS2.p2.1.m1.1.1.1.1.2.3.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1.1.1.2.3">1</cn></apply><apply id="S4.SS2.p2.1.m1.1.1.1.1.4.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.1.1.4.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.4">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.1.1.4.2.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.4.2">𝜆</ci><cn id="S4.SS2.p2.1.m1.1.1.1.1.4.3.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1.1.1.4.3">2</cn></apply></apply><apply id="S4.SS2.p2.1.m1.1.1.1.1c.cmml" xref="S4.SS2.p2.1.m1.1.1.1"><eq id="S4.SS2.p2.1.m1.1.1.1.1.5.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2411.14397v1#S4.SS2.p2.1.m1.1.1.1.1.4.cmml" id="S4.SS2.p2.1.m1.1.1.1.1d.cmml" xref="S4.SS2.p2.1.m1.1.1.1"></share><apply id="S4.SS2.p2.1.m1.1.1.1.1.6.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.6"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.1.1.6.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.6">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.1.1.6.2.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.6.2">𝜆</ci><cn id="S4.SS2.p2.1.m1.1.1.1.1.6.3.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1.1.1.6.3">3</cn></apply></apply><apply id="S4.SS2.p2.1.m1.1.1.1.1e.cmml" xref="S4.SS2.p2.1.m1.1.1.1"><eq id="S4.SS2.p2.1.m1.1.1.1.1.7.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.7"></eq><share href="https://arxiv.org/html/2411.14397v1#S4.SS2.p2.1.m1.1.1.1.1.6.cmml" id="S4.SS2.p2.1.m1.1.1.1.1f.cmml" xref="S4.SS2.p2.1.m1.1.1.1"></share><apply id="S4.SS2.p2.1.m1.1.1.1.1.8.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.8"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.1.1.8.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.8">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.1.1.8.2.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.8.2">𝜆</ci><cn id="S4.SS2.p2.1.m1.1.1.1.1.8.3.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1.1.1.8.3">4</cn></apply></apply><apply id="S4.SS2.p2.1.m1.1.1.1.1g.cmml" xref="S4.SS2.p2.1.m1.1.1.1"><eq id="S4.SS2.p2.1.m1.1.1.1.1.9.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.9"></eq><share href="https://arxiv.org/html/2411.14397v1#S4.SS2.p2.1.m1.1.1.1.1.8.cmml" id="S4.SS2.p2.1.m1.1.1.1.1h.cmml" xref="S4.SS2.p2.1.m1.1.1.1"></share><cn id="S4.SS2.p2.1.m1.1.1.1.1.10.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1.1.1.10">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">\lambda_{1}=\lambda_{2}=\lambda_{3}=\lambda_{4}=0.</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 0 .</annotation></semantics></math> Then the continuity conditions (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.E31" title="In 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">31</span></a>) and the current conservation (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.E32" title="In 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">32</span></a>) are for <math alttext="j=2,3,4" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.3"><semantics id="S4.SS2.p2.2.m2.3a"><mrow id="S4.SS2.p2.2.m2.3.4" xref="S4.SS2.p2.2.m2.3.4.cmml"><mi id="S4.SS2.p2.2.m2.3.4.2" xref="S4.SS2.p2.2.m2.3.4.2.cmml">j</mi><mo id="S4.SS2.p2.2.m2.3.4.1" xref="S4.SS2.p2.2.m2.3.4.1.cmml">=</mo><mrow id="S4.SS2.p2.2.m2.3.4.3.2" xref="S4.SS2.p2.2.m2.3.4.3.1.cmml"><mn id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">2</mn><mo id="S4.SS2.p2.2.m2.3.4.3.2.1" xref="S4.SS2.p2.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.SS2.p2.2.m2.2.2" xref="S4.SS2.p2.2.m2.2.2.cmml">3</mn><mo id="S4.SS2.p2.2.m2.3.4.3.2.2" xref="S4.SS2.p2.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.SS2.p2.2.m2.3.3" xref="S4.SS2.p2.2.m2.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.3b"><apply id="S4.SS2.p2.2.m2.3.4.cmml" xref="S4.SS2.p2.2.m2.3.4"><eq id="S4.SS2.p2.2.m2.3.4.1.cmml" xref="S4.SS2.p2.2.m2.3.4.1"></eq><ci id="S4.SS2.p2.2.m2.3.4.2.cmml" xref="S4.SS2.p2.2.m2.3.4.2">𝑗</ci><list id="S4.SS2.p2.2.m2.3.4.3.1.cmml" xref="S4.SS2.p2.2.m2.3.4.3.2"><cn id="S4.SS2.p2.2.m2.1.1.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1">2</cn><cn id="S4.SS2.p2.2.m2.2.2.cmml" type="integer" xref="S4.SS2.p2.2.m2.2.2">3</cn><cn id="S4.SS2.p2.2.m2.3.3.cmml" type="integer" xref="S4.SS2.p2.2.m2.3.3">4</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.3c">j=2,3,4</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.3d">italic_j = 2 , 3 , 4</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx30"> <tbody id="S4.E38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}&\Psi_{1,j}^{(a_{1,j})}(0)=\phi_{1},\quad\Psi_{1,j}^% {(a_{1,j})}(L_{1,j})=\phi_{j},\\ &\Psi_{1,j}^{(a_{1,j})}(L_{1,j})-\Psi_{1,j}^{(a_{1,j})}(L_{1,j}-a_{1,j})=0,\\ &\sum_{j=2}^{4}[\Psi_{1,j}^{(a_{1,j})}(a_{1,j})-\Psi_{1,j}^{(a_{1,j})}(0)]=0.% \end{split}" class="ltx_math_unparsed" display="inline" id="S4.E38.m1.67"><semantics id="S4.E38.m1.67a"><mtable columnspacing="0pt" id="S4.E38.m1.67.67.3" rowspacing="0pt"><mtr id="S4.E38.m1.67.67.3a"><mtd id="S4.E38.m1.67.67.3b"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E38.m1.67.67.3c"><mrow id="S4.E38.m1.65.65.1.65.22.22.22"><mrow id="S4.E38.m1.65.65.1.65.22.22.22.1"><mrow id="S4.E38.m1.65.65.1.65.22.22.22.1.1.1"><mrow id="S4.E38.m1.65.65.1.65.22.22.22.1.1.1.1"><msubsup id="S4.E38.m1.65.65.1.65.22.22.22.1.1.1.1.2"><mi id="S4.E38.m1.1.1.1.1.1.1" mathvariant="normal">Ψ</mi><mrow id="S4.E38.m1.2.2.2.2.2.2.1.4"><mn id="S4.E38.m1.2.2.2.2.2.2.1.1">1</mn><mo id="S4.E38.m1.2.2.2.2.2.2.1.4.1">,</mo><mi id="S4.E38.m1.2.2.2.2.2.2.1.2">j</mi></mrow><mrow id="S4.E38.m1.3.3.3.3.3.3.1.3"><mo id="S4.E38.m1.3.3.3.3.3.3.1.3.2" stretchy="false">(</mo><msub id="S4.E38.m1.3.3.3.3.3.3.1.3.1"><mi id="S4.E38.m1.3.3.3.3.3.3.1.3.1.2">a</mi><mrow id="S4.E38.m1.3.3.3.3.3.3.1.2.2.4"><mn id="S4.E38.m1.3.3.3.3.3.3.1.1.1.1">1</mn><mo id="S4.E38.m1.3.3.3.3.3.3.1.2.2.4.1">,</mo><mi id="S4.E38.m1.3.3.3.3.3.3.1.2.2.2">j</mi></mrow></msub><mo id="S4.E38.m1.3.3.3.3.3.3.1.3.3" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E38.m1.65.65.1.65.22.22.22.1.1.1.1.1">⁢</mo><mrow id="S4.E38.m1.65.65.1.65.22.22.22.1.1.1.1.3"><mo id="S4.E38.m1.4.4.4.4.4.4" stretchy="false">(</mo><mn id="S4.E38.m1.5.5.5.5.5.5">0</mn><mo id="S4.E38.m1.6.6.6.6.6.6" stretchy="false">)</mo></mrow></mrow><mo id="S4.E38.m1.7.7.7.7.7.7">=</mo><msub id="S4.E38.m1.65.65.1.65.22.22.22.1.1.1.2"><mi id="S4.E38.m1.8.8.8.8.8.8">ϕ</mi><mn id="S4.E38.m1.9.9.9.9.9.9.1">1</mn></msub></mrow><mo id="S4.E38.m1.10.10.10.10.10.10" rspace="1.167em">,</mo><mrow id="S4.E38.m1.65.65.1.65.22.22.22.1.2.2"><mrow id="S4.E38.m1.65.65.1.65.22.22.22.1.2.2.1"><msubsup id="S4.E38.m1.65.65.1.65.22.22.22.1.2.2.1.3"><mi id="S4.E38.m1.11.11.11.11.11.11" mathvariant="normal">Ψ</mi><mrow id="S4.E38.m1.12.12.12.12.12.12.1.4"><mn id="S4.E38.m1.12.12.12.12.12.12.1.1">1</mn><mo id="S4.E38.m1.12.12.12.12.12.12.1.4.1">,</mo><mi id="S4.E38.m1.12.12.12.12.12.12.1.2">j</mi></mrow><mrow id="S4.E38.m1.13.13.13.13.13.13.1.3"><mo id="S4.E38.m1.13.13.13.13.13.13.1.3.2" stretchy="false">(</mo><msub id="S4.E38.m1.13.13.13.13.13.13.1.3.1"><mi id="S4.E38.m1.13.13.13.13.13.13.1.3.1.2">a</mi><mrow id="S4.E38.m1.13.13.13.13.13.13.1.2.2.4"><mn id="S4.E38.m1.13.13.13.13.13.13.1.1.1.1">1</mn><mo id="S4.E38.m1.13.13.13.13.13.13.1.2.2.4.1">,</mo><mi id="S4.E38.m1.13.13.13.13.13.13.1.2.2.2">j</mi></mrow></msub><mo id="S4.E38.m1.13.13.13.13.13.13.1.3.3" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E38.m1.65.65.1.65.22.22.22.1.2.2.1.2">⁢</mo><mrow id="S4.E38.m1.65.65.1.65.22.22.22.1.2.2.1.1.1"><mo id="S4.E38.m1.14.14.14.14.14.14" stretchy="false">(</mo><msub id="S4.E38.m1.65.65.1.65.22.22.22.1.2.2.1.1.1.1"><mi id="S4.E38.m1.15.15.15.15.15.15">L</mi><mrow id="S4.E38.m1.16.16.16.16.16.16.1.4"><mn id="S4.E38.m1.16.16.16.16.16.16.1.1">1</mn><mo id="S4.E38.m1.16.16.16.16.16.16.1.4.1">,</mo><mi id="S4.E38.m1.16.16.16.16.16.16.1.2">j</mi></mrow></msub><mo id="S4.E38.m1.17.17.17.17.17.17" stretchy="false">)</mo></mrow></mrow><mo id="S4.E38.m1.18.18.18.18.18.18">=</mo><msub id="S4.E38.m1.65.65.1.65.22.22.22.1.2.2.2"><mi id="S4.E38.m1.19.19.19.19.19.19">ϕ</mi><mi id="S4.E38.m1.20.20.20.20.20.20.1">j</mi></msub></mrow></mrow><mo id="S4.E38.m1.21.21.21.21.21.21">,</mo></mrow></mtd></mtr><mtr id="S4.E38.m1.67.67.3d"><mtd id="S4.E38.m1.67.67.3e"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E38.m1.67.67.3f"><mrow id="S4.E38.m1.66.66.2.66.22.22.22"><mrow id="S4.E38.m1.66.66.2.66.22.22.22.1"><mrow id="S4.E38.m1.66.66.2.66.22.22.22.1.2"><mrow id="S4.E38.m1.66.66.2.66.22.22.22.1.1.1"><msubsup id="S4.E38.m1.66.66.2.66.22.22.22.1.1.1.3"><mi id="S4.E38.m1.22.22.22.1.1.1" mathvariant="normal">Ψ</mi><mrow id="S4.E38.m1.23.23.23.2.2.2.1.4"><mn id="S4.E38.m1.23.23.23.2.2.2.1.1">1</mn><mo id="S4.E38.m1.23.23.23.2.2.2.1.4.1">,</mo><mi id="S4.E38.m1.23.23.23.2.2.2.1.2">j</mi></mrow><mrow id="S4.E38.m1.24.24.24.3.3.3.1.3"><mo id="S4.E38.m1.24.24.24.3.3.3.1.3.2" 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id="S4.E38.m1.66.66.2.66.22.22.22.1.2.2"><msubsup id="S4.E38.m1.66.66.2.66.22.22.22.1.2.2.3"><mi id="S4.E38.m1.30.30.30.9.9.9" mathvariant="normal">Ψ</mi><mrow id="S4.E38.m1.31.31.31.10.10.10.1.4"><mn id="S4.E38.m1.31.31.31.10.10.10.1.1">1</mn><mo id="S4.E38.m1.31.31.31.10.10.10.1.4.1">,</mo><mi id="S4.E38.m1.31.31.31.10.10.10.1.2">j</mi></mrow><mrow id="S4.E38.m1.32.32.32.11.11.11.1.3"><mo id="S4.E38.m1.32.32.32.11.11.11.1.3.2" stretchy="false">(</mo><msub id="S4.E38.m1.32.32.32.11.11.11.1.3.1"><mi id="S4.E38.m1.32.32.32.11.11.11.1.3.1.2">a</mi><mrow id="S4.E38.m1.32.32.32.11.11.11.1.2.2.4"><mn id="S4.E38.m1.32.32.32.11.11.11.1.1.1.1">1</mn><mo id="S4.E38.m1.32.32.32.11.11.11.1.2.2.4.1">,</mo><mi id="S4.E38.m1.32.32.32.11.11.11.1.2.2.2">j</mi></mrow></msub><mo id="S4.E38.m1.32.32.32.11.11.11.1.3.3" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E38.m1.66.66.2.66.22.22.22.1.2.2.2">⁢</mo><mrow id="S4.E38.m1.66.66.2.66.22.22.22.1.2.2.1.1"><mo id="S4.E38.m1.33.33.33.12.12.12" 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id="S4.E38.m1.67.67.3h"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E38.m1.67.67.3i"><mrow id="S4.E38.m1.67.67.3.67.23.23.23"><mrow id="S4.E38.m1.67.67.3.67.23.23.23.1"><mrow id="S4.E38.m1.67.67.3.67.23.23.23.1.1"><mstyle displaystyle="true" id="S4.E38.m1.67.67.3.67.23.23.23.1.1.2"><munderover id="S4.E38.m1.67.67.3.67.23.23.23.1.1.2a"><mo id="S4.E38.m1.43.43.43.1.1.1" movablelimits="false">∑</mo><mrow id="S4.E38.m1.44.44.44.2.2.2.1"><mi id="S4.E38.m1.44.44.44.2.2.2.1.2">j</mi><mo id="S4.E38.m1.44.44.44.2.2.2.1.1">=</mo><mn id="S4.E38.m1.44.44.44.2.2.2.1.3">2</mn></mrow><mn id="S4.E38.m1.45.45.45.3.3.3.1">4</mn></munderover></mstyle><mrow id="S4.E38.m1.67.67.3.67.23.23.23.1.1.1.1"><mo id="S4.E38.m1.46.46.46.4.4.4" stretchy="false">[</mo><mrow id="S4.E38.m1.67.67.3.67.23.23.23.1.1.1.1.1"><mrow id="S4.E38.m1.67.67.3.67.23.23.23.1.1.1.1.1.1"><msubsup id="S4.E38.m1.67.67.3.67.23.23.23.1.1.1.1.1.1.3"><mi id="S4.E38.m1.47.47.47.5.5.5" mathvariant="normal">Ψ</mi><mrow 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&\sum_{j=2}^{4}[\Psi_{1,j}^{(a_{1,j})}(a_{1,j})-\Psi_{1,j}^{(a_{1,j})}(0)]=0.% \end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E38.m1.67c">start_ROW start_CELL end_CELL start_CELL roman_Ψ start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( 0 ) = italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_Ψ start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL roman_Ψ start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) - roman_Ψ start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) = 0 , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_j = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT [ roman_Ψ start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) - roman_Ψ start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( 0 ) ] = 0 . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(38)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.1">Substitution of the solution in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.Ex21" title="4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">4.1</span></a>) into (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.E38" title="In 4.2 Star branched lattice ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">38</span></a>) yields the secular equations</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx31"> <tbody id="S4.Ex28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle A_{1,j}f_{1,j}(N_{1,j})=\phi_{1},\quad B_{1,j}f_{1,j}(N_{1,j})=% \phi_{j}," class="ltx_Math" display="inline" 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xref="S4.Ex28.m1.13.13.1.1.2.2.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex28.m1.13c">\displaystyle A_{1,j}f_{1,j}(N_{1,j})=\phi_{1},\quad B_{1,j}f_{1,j}(N_{1,j})=% \phi_{j},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex28.m1.13d">italic_A start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) ] = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(39)</span></td> </tr></tbody> <tbody id="S4.Ex29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{j=2}^{4}\left\{A_{1,j}\left[f_{1,j}(N_{1,j}-1)-f_{1,j}(N_{1% ,j})\right]+B_{1,j}f_{1,j}(1)\right\}=0." class="ltx_Math" display="inline" id="S4.Ex29.m1.16"><semantics id="S4.Ex29.m1.16a"><mrow id="S4.Ex29.m1.16.16.1" xref="S4.Ex29.m1.16.16.1.1.cmml"><mrow id="S4.Ex29.m1.16.16.1.1" xref="S4.Ex29.m1.16.16.1.1.cmml"><mrow id="S4.Ex29.m1.16.16.1.1.1" xref="S4.Ex29.m1.16.16.1.1.1.cmml"><mstyle 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xref="S4.Ex29.m1.16.16.1.1.1.1.1.1.3.3.2">𝑓</ci><list id="S4.Ex29.m1.14.14.2.3.cmml" xref="S4.Ex29.m1.14.14.2.4"><cn id="S4.Ex29.m1.13.13.1.1.cmml" type="integer" xref="S4.Ex29.m1.13.13.1.1">1</cn><ci id="S4.Ex29.m1.14.14.2.2.cmml" xref="S4.Ex29.m1.14.14.2.2">𝑗</ci></list></apply><cn id="S4.Ex29.m1.15.15.cmml" type="integer" xref="S4.Ex29.m1.15.15">1</cn></apply></apply></set></apply><cn id="S4.Ex29.m1.16.16.1.1.3.cmml" type="integer" xref="S4.Ex29.m1.16.16.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex29.m1.16c">\displaystyle\sum_{j=2}^{4}\left\{A_{1,j}\left[f_{1,j}(N_{1,j}-1)-f_{1,j}(N_{1% ,j})\right]+B_{1,j}f_{1,j}(1)\right\}=0.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex29.m1.16d">∑ start_POSTSUBSCRIPT italic_j = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT { italic_A start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT [ italic_f start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT - 1 ) - italic_f start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) ] + italic_B start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( 1 ) } = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.6">The above system of equations has a non-trivial solution, when</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx32"> <tbody id="S4.E40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\det{(M(k))}=0," class="ltx_Math" display="inline" id="S4.E40.m1.2"><semantics id="S4.E40.m1.2a"><mrow id="S4.E40.m1.2.2.1" 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xref="S4.E40.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E40.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.E40.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E40.m1.2.2.1.1.2" xref="S4.E40.m1.2.2.1.1.2.cmml">=</mo><mn id="S4.E40.m1.2.2.1.1.3" xref="S4.E40.m1.2.2.1.1.3.cmml">0</mn></mrow><mo id="S4.E40.m1.2.2.1.2" xref="S4.E40.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E40.m1.2b"><apply id="S4.E40.m1.2.2.1.1.cmml" xref="S4.E40.m1.2.2.1"><eq id="S4.E40.m1.2.2.1.1.2.cmml" xref="S4.E40.m1.2.2.1.1.2"></eq><apply id="S4.E40.m1.2.2.1.1.1.cmml" xref="S4.E40.m1.2.2.1.1.1"><determinant id="S4.E40.m1.2.2.1.1.1.2.cmml" xref="S4.E40.m1.2.2.1.1.1.2"></determinant><apply id="S4.E40.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E40.m1.2.2.1.1.1.1.1"><times id="S4.E40.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E40.m1.2.2.1.1.1.1.1.1.1"></times><ci id="S4.E40.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E40.m1.2.2.1.1.1.1.1.1.2">𝑀</ci><ci id="S4.E40.m1.1.1.cmml" xref="S4.E40.m1.1.1">𝑘</ci></apply></apply><cn id="S4.E40.m1.2.2.1.1.3.cmml" type="integer" xref="S4.E40.m1.2.2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E40.m1.2c">\displaystyle\det{(M(k))}=0,</annotation><annotation encoding="application/x-llamapun" id="S4.E40.m1.2d">roman_det ( italic_M ( italic_k ) ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(40)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p4.1">where the <math alttext="M(k)" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><mrow id="S4.SS2.p4.1.m1.1.2" xref="S4.SS2.p4.1.m1.1.2.cmml"><mi id="S4.SS2.p4.1.m1.1.2.2" xref="S4.SS2.p4.1.m1.1.2.2.cmml">M</mi><mo id="S4.SS2.p4.1.m1.1.2.1" xref="S4.SS2.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.1.m1.1.2.3.2" xref="S4.SS2.p4.1.m1.1.2.cmml"><mo id="S4.SS2.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.1.m1.1.2.cmml">(</mo><mi id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml">k</mi><mo id="S4.SS2.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.2"><times id="S4.SS2.p4.1.m1.1.2.1.cmml" xref="S4.SS2.p4.1.m1.1.2.1"></times><ci id="S4.SS2.p4.1.m1.1.2.2.cmml" xref="S4.SS2.p4.1.m1.1.2.2">𝑀</ci><ci id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">M(k)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">italic_M ( italic_k )</annotation></semantics></math> matrix is the same as the continuous version of the star graph</p> <table class="ltx_equationgroup ltx_eqn_eqnarray 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xref="S4.E41.m1.1.1.1.1.8.8.4.2.4.2">𝑈</ci><cn id="S4.E41.m1.1.1.1.1.7.7.3.1.1.1.1.cmml" type="integer" xref="S4.E41.m1.1.1.1.1.7.7.3.1.1.1.1">2</cn></apply><ci id="S4.E41.m1.1.1.1.1.8.8.4.2.2.cmml" xref="S4.E41.m1.1.1.1.1.8.8.4.2.2">𝑘</ci></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E41.m1.3c">\displaystyle M(k)=\begin{pmatrix}\Lambda^{(1)}&W(k)&0\\ \Lambda^{(2)}&0&W(k)\\ 0&U^{(1)}(k)&U^{(2)}(k)\end{pmatrix}.</annotation><annotation encoding="application/x-llamapun" id="S4.E41.m1.3d">italic_M ( italic_k ) = ( start_ARG start_ROW start_CELL roman_Λ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT end_CELL start_CELL italic_W ( italic_k ) end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL roman_Λ start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_W ( italic_k ) end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_U start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_k ) end_CELL start_CELL italic_U start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_k ) end_CELL end_ROW end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(41)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p4.7">The elements of the above matrices are</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx34"> <tbody id="S4.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\Lambda^{(1)}=\begin{pmatrix}-1&0&0&0\\ -1&0&0&0\\ -1&0&0&0\end{pmatrix},\quad\Lambda^{(2)}=\begin{pmatrix}0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}," class="ltx_Math" display="inline" id="S4.Ex30.m1.5"><semantics 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id="S4.Ex30.m1.5d">roman_Λ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL - 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL - 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL - 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW end_ARG ) , roman_Λ start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL 0 end_CELL start_CELL - 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL - 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL - 1 end_CELL end_ROW end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex31"><tr class="ltx_equation ltx_eqn_row 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W(k)=\begin{pmatrix}f_{1,2}(N_{1,2})&0&0\\ 0&f_{1,3}(N_{1,3})&0\\ 0&0&f_{1,4}(N_{1,4})\end{pmatrix},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex31.m1.3d">italic_W ( italic_k ) = ( start_ARG start_ROW start_CELL italic_f start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT ) end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_f start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ) end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_f start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ) end_CELL end_ROW end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ) end_CELL start_CELL italic_r start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL italic_f start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT ( 1 ) end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_f start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ( 1 ) end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_f start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ( 1 ) end_CELL end_ROW end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 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xref="S4.Ex33.m1.1.1.1.1.23.23.4.4.4.2.2">4</cn></list></apply></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex33.m1.4c">\displaystyle U^{(2)}(k)=\begin{pmatrix}f_{1,2}(1)&f_{1,3}(1)&f_{1,4}(1)\\ r_{1,2}(N_{1,2})&0&0\\ 0&r_{1,3}(N_{1,3})&0\\ 0&0&r_{1,4}(N_{1,4})\end{pmatrix},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex33.m1.4d">italic_U start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_k ) = ( start_ARG start_ROW start_CELL italic_f start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT ( 1 ) end_CELL start_CELL italic_f start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ( 1 ) end_CELL start_CELL italic_f start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ( 1 ) end_CELL end_ROW start_ROW start_CELL italic_r start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT ) end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_r start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT ) end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_r start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT ) end_CELL end_ROW end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p4.5">where <math alttext="r_{1,j}(N_{1,j})=f_{1,j}(N_{1,j}-1)-f_{1,j}(N_{1,j})" class="ltx_Math" display="inline" id="S4.SS2.p4.2.m1.15"><semantics id="S4.SS2.p4.2.m1.15a"><mrow id="S4.SS2.p4.2.m1.15.15" xref="S4.SS2.p4.2.m1.15.15.cmml"><mrow id="S4.SS2.p4.2.m1.13.13.1" xref="S4.SS2.p4.2.m1.13.13.1.cmml"><msub id="S4.SS2.p4.2.m1.13.13.1.3" xref="S4.SS2.p4.2.m1.13.13.1.3.cmml"><mi id="S4.SS2.p4.2.m1.13.13.1.3.2" xref="S4.SS2.p4.2.m1.13.13.1.3.2.cmml">r</mi><mrow id="S4.SS2.p4.2.m1.2.2.2.4" 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xref="S4.SS2.p4.2.m1.12.12.2.2">𝑗</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.2.m1.15c">r_{1,j}(N_{1,j})=f_{1,j}(N_{1,j}-1)-f_{1,j}(N_{1,j})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.2.m1.15d">italic_r start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT - 1 ) - italic_f start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT )</annotation></semantics></math>. This secular equation can be solved numerically to provide eigenvalues of discrete quantum star graph. Here one should note that if the lengths of the edges are equal the eigenvalue spectrum becomes degenerate, i.e. has multiple eigenvalues. To avoid such situation, when we solve the secular equation, we choose lengths of the edges as rationally independent. The first five nonzero eigenvalues are calculated for the considered star graph (see Table <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.T2" title="Table 2 ‣ 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a>), where the step size for each edge of the graph is chosen to be equal, <math alttext="a_{1,j}=a" class="ltx_Math" display="inline" id="S4.SS2.p4.3.m2.2"><semantics id="S4.SS2.p4.3.m2.2a"><mrow id="S4.SS2.p4.3.m2.2.3" xref="S4.SS2.p4.3.m2.2.3.cmml"><msub id="S4.SS2.p4.3.m2.2.3.2" xref="S4.SS2.p4.3.m2.2.3.2.cmml"><mi id="S4.SS2.p4.3.m2.2.3.2.2" xref="S4.SS2.p4.3.m2.2.3.2.2.cmml">a</mi><mrow id="S4.SS2.p4.3.m2.2.2.2.4" xref="S4.SS2.p4.3.m2.2.2.2.3.cmml"><mn id="S4.SS2.p4.3.m2.1.1.1.1" xref="S4.SS2.p4.3.m2.1.1.1.1.cmml">1</mn><mo id="S4.SS2.p4.3.m2.2.2.2.4.1" xref="S4.SS2.p4.3.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p4.3.m2.2.2.2.2" xref="S4.SS2.p4.3.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS2.p4.3.m2.2.3.1" xref="S4.SS2.p4.3.m2.2.3.1.cmml">=</mo><mi id="S4.SS2.p4.3.m2.2.3.3" xref="S4.SS2.p4.3.m2.2.3.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.3.m2.2b"><apply id="S4.SS2.p4.3.m2.2.3.cmml" xref="S4.SS2.p4.3.m2.2.3"><eq id="S4.SS2.p4.3.m2.2.3.1.cmml" xref="S4.SS2.p4.3.m2.2.3.1"></eq><apply id="S4.SS2.p4.3.m2.2.3.2.cmml" xref="S4.SS2.p4.3.m2.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p4.3.m2.2.3.2.1.cmml" xref="S4.SS2.p4.3.m2.2.3.2">subscript</csymbol><ci id="S4.SS2.p4.3.m2.2.3.2.2.cmml" xref="S4.SS2.p4.3.m2.2.3.2.2">𝑎</ci><list id="S4.SS2.p4.3.m2.2.2.2.3.cmml" xref="S4.SS2.p4.3.m2.2.2.2.4"><cn id="S4.SS2.p4.3.m2.1.1.1.1.cmml" type="integer" xref="S4.SS2.p4.3.m2.1.1.1.1">1</cn><ci id="S4.SS2.p4.3.m2.2.2.2.2.cmml" xref="S4.SS2.p4.3.m2.2.2.2.2">𝑗</ci></list></apply><ci id="S4.SS2.p4.3.m2.2.3.3.cmml" xref="S4.SS2.p4.3.m2.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.3.m2.2c">a_{1,j}=a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.3.m2.2d">italic_a start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT = italic_a</annotation></semantics></math> and lengths of edges of the graph are chosen as <math alttext="L_{1,2}=0.8,\;L_{1,3}=1.1,\;L_{1,4}=1.5" class="ltx_Math" display="inline" id="S4.SS2.p4.4.m3.8"><semantics id="S4.SS2.p4.4.m3.8a"><mrow id="S4.SS2.p4.4.m3.8.8.2" xref="S4.SS2.p4.4.m3.8.8.3.cmml"><mrow id="S4.SS2.p4.4.m3.7.7.1.1" xref="S4.SS2.p4.4.m3.7.7.1.1.cmml"><msub id="S4.SS2.p4.4.m3.7.7.1.1.2" xref="S4.SS2.p4.4.m3.7.7.1.1.2.cmml"><mi id="S4.SS2.p4.4.m3.7.7.1.1.2.2" xref="S4.SS2.p4.4.m3.7.7.1.1.2.2.cmml">L</mi><mrow id="S4.SS2.p4.4.m3.2.2.2.4" xref="S4.SS2.p4.4.m3.2.2.2.3.cmml"><mn id="S4.SS2.p4.4.m3.1.1.1.1" xref="S4.SS2.p4.4.m3.1.1.1.1.cmml">1</mn><mo id="S4.SS2.p4.4.m3.2.2.2.4.1" 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xref="S4.SS2.p4.4.m3.8.8.2.2.2.2.3">1.5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.4.m3.8c">L_{1,2}=0.8,\;L_{1,3}=1.1,\;L_{1,4}=1.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.4.m3.8d">italic_L start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = 0.8 , italic_L start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT = 1.1 , italic_L start_POSTSUBSCRIPT 1 , 4 end_POSTSUBSCRIPT = 1.5</annotation></semantics></math>. In each column (2-6 columns) in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.T2" title="Table 2 ‣ 4.1 Arbitrary branching topology ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a> we present nonzero eigenvalues by choosing the different values of step size <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.p4.5.m4.1"><semantics id="S4.SS2.p4.5.m4.1a"><mi id="S4.SS2.p4.5.m4.1.1" xref="S4.SS2.p4.5.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.5.m4.1b"><ci id="S4.SS2.p4.5.m4.1.1.cmml" xref="S4.SS2.p4.5.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.5.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.5.m4.1d">italic_a</annotation></semantics></math> for the star graph. The results show the convergence of solutions to the solutions of the continuous case calculated using Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S3.E27" title="In 3 Basic theory of quantum graphs ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">27</span></a>) for the continuous star graph for the same lengths of edges.</p> </div> <figure class="ltx_figure" id="S4.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F3.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="57" id="S4.F3.sf1.g1" src="x3.png" width="381"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(a) </span>Linear (unbranched) conducting polymer lattice.</figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F3.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="225" id="S4.F3.sf2.g1" src="x4.png" width="381"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(b) </span>Branched conducting polymer.</figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Conducting polymer chains.</figcaption> </figure> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Possible experimental realization of the model</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">Here we briefly discuss the possible practical application of our model to so-called conducting polymers. These latter are the pol-conjugated (polymer) molecular chains exhibiting semiconducting electronic properties and therefore are called organic semiconductors. Conducting polymers are the basic functional materials for organic electronics and so-called polymer based film organic photovoltaics (see e.g., Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib36" title="">36</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib39" title="">39</a>]</cite>, for review of conducting polymers and their applications). The structure of conducting polymers presents a discrete lattice (chain) with the hexagonal basic cell (see, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.F3" title="Figure 3 ‣ 4.2 Star branched lattice ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">3</span></a>a). Charge carrier dynamics in such structure can be described in terms of the discrete Schrödinger equation given by Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S2.E2" title="In 2 Exact solution and spectrum of discrete Schrödinger equation on a finite chain ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">2</span></a>). An important problem arising in the context of charge dynamics in conducting polymers is modeling electron or exciton (in some cases polarons and solitons) transport with account of its discrete structure. The solution of such a problem allows us to compute the electronic band structure of the material and tuning of its electronic properties. The latter is important for functional optimization of conducting polymers and improving of device performance in organic photovoltaics. Recently, the structures consisting of branched conducting polymer chains attracted much attention in the context of polymer based photovoltaics and organic electronics (see, e.g. Refs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib39" title="">39</a>]</cite> for review). In the simplest case, such polymer can have a star-branched structure (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#S4.F3" title="Figure 3 ‣ 4.2 Star branched lattice ‣ 4 Branched lattices ‣ Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice"><span class="ltx_text ltx_ref_tag">3</span></a>b). Our model for branched quantum lattice based on the use of discrete Schrödinger equation on the metric graph can be effectively used for the description of charge carrier dynamics and computation of the electronic band spectrum in branched conducting polymers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib34" title="">34</a>]</cite>. Also, charge current, and current-voltage characteristics in the AC driven case can be computed. Another possible application can be using DSE for modeling of so-called coherent Ising machines, which can be realized using the network states <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib41" title="">41</a>]</cite>. Also the model we proposed can be used for description of chip-based photonic graph states <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib42" title="">42</a>]</cite> and holonomic quantum gates <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14397v1#bib.bib43" title="">43</a>]</cite>.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Conclusion</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this paper, we considered the problem of discrete Schrödinger equation on a metric graph, by focusing on its analytical solution and continuum limit. An exact solution of DSE on a finite interval is obtained. Consistency of the result with their continuum counterpart is shown by explicit calculation of the continuum limit of the solution. The result is extended to the case of quantum graph of arbitrary topology. Application of the obtained result to the special case of quantum star graph is demonstrated by imposing the vertex matching conditions given in the form of continuity and Kirchhoff rule. It is shown that the secular equation derived for star graph reproduces its well-know continuum counterpart. The practical application of the proposed model to branched conducting polymers is discussed.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Acknowledgements</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">This work is supported by European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement ID: 873071, project SOMPATY. The work of JY and DM is partially supported by the grant of the Innovation Development Agency of the Republic of Uzbekistan (Ref. 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