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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen</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/2503.15710v1/"/></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/2503.15710v1#S1" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S2" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span>Hydrogen in Parallel Electric and Magnetic Fields</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S3" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span>Escape Rate from Classical Trajectory Monte Carlo</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S4" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span>Periodic Orbit Theory and Spectral Determinants</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S5" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span>Surface of Section and Discrete-Time Monte Carlo</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span>Computing Periodic Orbits via Phase Space Partitioning</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VII </span>Escape Rate from Periodic Orbits</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VIII </span>Locating Hyperbolic Plateaus</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S9" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IX </span>Varying the electron energy</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S10" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">X </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S11" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">XI </span>Author Declarations</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/2503.15710v1#S11.SS1" title="In XI Author Declarations ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">XI.1 </span>Conflict of Interest</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S11.SS2" title="In XI Author Declarations ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">XI.2 </span>Author Contributions</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S12" title="In Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">XII </span>Data Availability</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line" lang="en"> <h1 class="ltx_title ltx_title_document">Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ethan T. Custodio </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:ecustodio@ucmerced.edu">ecustodio@ucmerced.edu</a> </span> <span class="ltx_contact ltx_role_affiliation">Physics Department, University of California, Merced, CA 95344, USA </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Sulimon Sattari </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 0010020, Japan </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Kevin A. Mitchell </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:kmitchell@ucmerced.edu">kmitchell@ucmerced.edu</a> </span> <span class="ltx_contact ltx_role_affiliation">Physics Department, University of California, Merced, CA 95344, USA </span></span></span> </div> <div class="ltx_dates">(<span class="ltx_text" id="id1.id1">March 19, 2025</span>)</div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id2.id1"><span class="ltx_text" id="id2.id1.1">When placed in parallel magnetic and electric fields, the electron trajectories of a classical hydrogen atom are chaotic. The classical escape rate of such a system can be computed with classical trajectory Monte Carlo techniques, but these computations require enormous numbers of trajectories, provide little understanding of the dynamical mechanisms involved, and must be completely rerun for any change of system parameter, no matter how small. We demonstrate an alternative technique to classical trajectory Monte Carlo computations, based on classical periodic orbit theory. In this technique, escape rates are computed from a relatively modest number (a few thousand) of periodic orbits of the system. One only needs the orbits’ periods and stability eigenvalues. A major advantage of this approach is that one does not need to repeat the entire analysis from scratch as system parameters are varied; one can numerically continue the periodic orbits instead. We demonstrate the periodic orbit technique for the ionization of a hydrogen atom in applied parallel electric and magnetic fields. Using fundamental theories of phase space geometry, we also show how to generate nontrivial symbolic dynamics for acquiring periodic orbits in physical systems. A detailed analysis of heteroclinic tangles and how they relate to bifurcations in periodic orbits is also presented.</span></p> </div> <div class="ltx_para" id="p1"> <blockquote class="ltx_quote" id="p1.1"> <p class="ltx_p" id="p1.1.1">Periodic orbits are special trajectories in nonlinear systems that form closed cycles. That is, after some finite time the trajectory will begin to retrace itself. Unstable periodic orbits have a neighborhood around them over which they exert some dynamical influence quantified by their Lyapunov exponent. Any trajectory in the system can be thought of as moving from the neighborhood of one unstable periodic orbit to another until it becomes trapped or escapes to infinity. Thus one can use these orbits as a “skeleton” to compute dynamical averages without the need for statistical simulation techniques like Monte Carlo methods. Here we will use the method of periodic orbits to compute escape rates based on a classical atomic Hamiltonian.</p> </blockquote> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">The importance of periodic orbits for characterizing classical chaotic dynamics was first realized by Poincaré <cite class="ltx_cite ltx_citemacro_cite">Poincaré (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib1" title="">1890</a>)</cite>, who understood periodic orbits as a skeleton of the phase space dynamics <cite class="ltx_cite ltx_citemacro_cite">Cvitanovic (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib2" title="">1991</a>)</cite>. Since Poincaré, a rich and well developed theory has developed that shows how periodic orbits can be used to compute and characterize the statistical behavior of classical and quantum dynamical systems <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib3" title="">Cvitanovic <em class="ltx_emph ltx_font_italic">et al.</em> </a></cite>. The appeal of such periodic orbit theory is that it reduces a complex system to a set of prototypical dynamical behaviors (the periodic orbits). The quantum problem has perhaps received the most attention. Gutzwiller, in his seminal work <cite class="ltx_cite ltx_citemacro_cite">Gutzwiller (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib4" title="">1971</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib5" title="">1990</a>)</cite>, showed that fluctuations in the spectral density of a quantum system were attributable to periodic orbits of the corresponding classical system. This ultimately led to significant new insights into the role periodic orbits play in physical applications, such as the absorption spectra of highly excited atoms (i.e. Rydberg atoms) in applied fields <cite class="ltx_cite ltx_citemacro_cite">Holle <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib6" title="">1986</a>); Main <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib7" title="">1986</a>); Holle <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib8" title="">1988</a>); Du and Delos (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib9" title="">1988a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib10" title="">b</a>); Main <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib11" title="">1994</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Though most applications of periodic orbit theory in quantum systems have looked at oscillations in the density of states, in a few select cases, periodic orbit theory could be pushed further to resolve energy levels (or resonances) of chaotic spectra as a sum over contributions from many periodic orbits <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib3" title="">Cvitanovic <em class="ltx_emph ltx_font_italic">et al.</em> </a></cite>. To date, we know of two well developed examples of this: the three-disk scatterer <cite class="ltx_cite ltx_citemacro_cite">Cvitanović and Eckhardt (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib12" title="">1989</a>)</cite> and the one-dimensional helium atom <cite class="ltx_cite ltx_citemacro_cite">Wintgen, Richter, and Tanner (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib13" title="">1992</a>)</cite>. The absence of more examples highlights a lack of understanding of the conditions under which periodic orbit theory can be successfully applied to resolve chaotic spectra.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">In addition to quantum applications, classical phase space averages can be computed from sums over periodic orbits. Again, the three-disk scatterer is a prototypical example, in which the escape rate of trajectories trapped between the three disks can be computed from a sum over periodic orbit contributions <cite class="ltx_cite ltx_citemacro_cite">Eckhardt <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib14" title="">1994</a>)</cite>. But there are still unresolved issues regarding how broadly such techniques can be applied. For example, despite some prior work <cite class="ltx_cite ltx_citemacro_cite">Sattari and Mitchell (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib15" title="">2017</a>)</cite>, the question of how to apply periodic orbit theory to mixed phase spaces remains an open challenge.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">There has recently also been a surge of interest in periodic orbit theory applied to high-dimensional phase spaces and dynamical systems defined by partial differential equations. Most notable here is the success in understanding the transition to fluid turbulence via periodic orbit decompositions of solutions to the Navier-Stokes equation at intermediate Reynolds number<cite class="ltx_cite ltx_citemacro_cite">Gibson, Halcrow, and Cvitanovic (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib16" title="">2008</a>); Budanur <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib17" title="">2017</a>); Graham and Floryan (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib18" title="">2021</a>); Avila, Barkley, and Hof (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib19" title="">2023</a>)</cite>. The hope is to eventually compute statistical averages of turbulent motion with a small number of periodic orbits capturing the essential features of turbulence. Higher dimensional turbulent systems rarely have a finite grammar which completely describes phase space transport. Thus, computing a full set of periodic orbits in such a system is an impossible task. In these systems a truncated set of periodic orbits computed based on stability can be used to compute dynamical averages<cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib3" title="">Cvitanovic <em class="ltx_emph ltx_font_italic">et al.</em> </a></cite>.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">As exciting as the recent high-dimensional developments are, there are still relatively few low-dimensional physical examples of the application of periodic orbit theory, especially those with quantitatively accurate periodic orbit computations. To help fill in this void, the current paper presents a highly accurate periodic orbit computation for the classical decay of a hydrogen atom in parallel electric and magnetic fields. We use the theory of heteroclinic tangles to create a Markov partition of phase space that completely describes phase space transport. Using this partition, we compute a complete set of periodic orbits up to a given discrete-time period. Then using the spectral determinant form of periodic orbit theory, we compute the classical decay rate across a wide range of parameter values, which span so-called hyperbolic plateaus. Utilizing the theory of heteroclinic tangles we are able to identify and compute the boundaries of two hyperbolic plateaus. The transition from one plateau to the other requires pruning of periodic orbits as the symbolic dynamics of the system changes.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">While this paper presents a relatively simple case with only two fixed points and a modest Hamiltonian, these methods can also be applied to more complicated two-dimensional systems. Homotopic Lobe Dynamics (HLD) can be used to partition phases with any number of fixed points and nested tangles so long as there is a finite symbolic dynamics<cite class="ltx_cite ltx_citemacro_cite">Mitchell (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib20" title="">2012</a>)</cite>. As system complexity increases additional a-priori knowledge becomes necessary to locate all the fixed points and find parameter values that have finite symbolic dynamics. The time it takes to compute all periodic orbits will certainly increase with complexity, but they only need to be computed once and then can be numerically continued through parameter space.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">The paper is organized as follows. In Sect. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S2" title="II Hydrogen in Parallel Electric and Magnetic Fields ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">II</span></a> the system and its Hamiltonian are introduced. In Sect. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S3" title="III Escape Rate from Classical Trajectory Monte Carlo ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">III</span></a> we compute the escape rate from a classical trajectory Monte Carlo simulation. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S4" title="IV Periodic Orbit Theory and Spectral Determinants ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">IV</span></a> presents a discussion of periodic orbit theory and how spectral determinants can be used to compute the escape rate. At this point the discrete dynamics are introduced. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S5" title="V Surface of Section and Discrete-Time Monte Carlo ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">V</span></a> defines a surface of section in order to define a discrete-time map. In Sect. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6" title="VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">VI</span></a> a method for computing periodic orbits is presented by creating Markov partitions of phase space using heteroclinic tangles. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7" title="VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">VII</span></a> uses the periodic orbits from the previous section to compute the escape rate with spectral determinants. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8" title="VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">VIII</span></a> discusses heteroclinic tangles in more detail and how their topology changes as parameters are varied. Two hyperbolic plateaus are identified using heteroclinic tangencies to define the borders. Finally, in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S9" title="IX Varying the electron energy ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">IX</span></a> we numerically continue the periodic orbits to fully explore both hyperbolic plateaus.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span>Hydrogen in Parallel Electric and Magnetic Fields</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">We consider hydrogen in parallel electric and magnetic fields because it retains an axis of symmetry, which reduces it to a two degree-of-freedom Hamiltonian system with chaos. 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xref="S2.E1.m1.3.3.1.1.3.4.3">superscript</csymbol><ci id="S2.E1.m1.3.3.1.1.3.4.3.2.cmml" xref="S2.E1.m1.3.3.1.1.3.4.3.2">𝐵</ci><cn id="S2.E1.m1.3.3.1.1.3.4.3.3.cmml" type="integer" xref="S2.E1.m1.3.3.1.1.3.4.3.3">2</cn></apply><apply id="S2.E1.m1.3.3.1.1.3.4.4.cmml" xref="S2.E1.m1.3.3.1.1.3.4.4"><csymbol cd="ambiguous" id="S2.E1.m1.3.3.1.1.3.4.4.1.cmml" xref="S2.E1.m1.3.3.1.1.3.4.4">superscript</csymbol><ci id="S2.E1.m1.3.3.1.1.3.4.4.2.cmml" xref="S2.E1.m1.3.3.1.1.3.4.4.2">𝜌</ci><cn id="S2.E1.m1.3.3.1.1.3.4.4.3.cmml" type="integer" xref="S2.E1.m1.3.3.1.1.3.4.4.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.3c">H\left(\rho,z,p_{\rho},p_{z}\right)=\frac{1}{2}\left(p_{\rho}^{2}+p_{z}^{2}% \right)-\frac{1}{\sqrt{\rho^{2}+z^{2}}}+Fz+\frac{1}{8}B^{2}\rho^{2},</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.3d">italic_H ( italic_ρ , italic_z , italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG + italic_F italic_z + divide start_ARG 1 end_ARG start_ARG 8 end_ARG italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 2 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">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.14">where <math alttext="(\rho,z)" class="ltx_Math" display="inline" id="S2.p1.2.m1.2"><semantics id="S2.p1.2.m1.2a"><mrow id="S2.p1.2.m1.2.3.2" xref="S2.p1.2.m1.2.3.1.cmml"><mo id="S2.p1.2.m1.2.3.2.1" stretchy="false" xref="S2.p1.2.m1.2.3.1.cmml">(</mo><mi id="S2.p1.2.m1.1.1" xref="S2.p1.2.m1.1.1.cmml">ρ</mi><mo id="S2.p1.2.m1.2.3.2.2" xref="S2.p1.2.m1.2.3.1.cmml">,</mo><mi id="S2.p1.2.m1.2.2" xref="S2.p1.2.m1.2.2.cmml">z</mi><mo id="S2.p1.2.m1.2.3.2.3" stretchy="false" xref="S2.p1.2.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m1.2b"><interval closure="open" id="S2.p1.2.m1.2.3.1.cmml" xref="S2.p1.2.m1.2.3.2"><ci id="S2.p1.2.m1.1.1.cmml" xref="S2.p1.2.m1.1.1">𝜌</ci><ci id="S2.p1.2.m1.2.2.cmml" xref="S2.p1.2.m1.2.2">𝑧</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m1.2c">(\rho,z)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m1.2d">( italic_ρ , italic_z )</annotation></semantics></math> are cylindrical coordinates and <math alttext="(p_{\rho},p_{z})" class="ltx_Math" display="inline" id="S2.p1.3.m2.2"><semantics id="S2.p1.3.m2.2a"><mrow id="S2.p1.3.m2.2.2.2" xref="S2.p1.3.m2.2.2.3.cmml"><mo id="S2.p1.3.m2.2.2.2.3" stretchy="false" xref="S2.p1.3.m2.2.2.3.cmml">(</mo><msub id="S2.p1.3.m2.1.1.1.1" xref="S2.p1.3.m2.1.1.1.1.cmml"><mi id="S2.p1.3.m2.1.1.1.1.2" xref="S2.p1.3.m2.1.1.1.1.2.cmml">p</mi><mi id="S2.p1.3.m2.1.1.1.1.3" xref="S2.p1.3.m2.1.1.1.1.3.cmml">ρ</mi></msub><mo id="S2.p1.3.m2.2.2.2.4" xref="S2.p1.3.m2.2.2.3.cmml">,</mo><msub id="S2.p1.3.m2.2.2.2.2" xref="S2.p1.3.m2.2.2.2.2.cmml"><mi id="S2.p1.3.m2.2.2.2.2.2" xref="S2.p1.3.m2.2.2.2.2.2.cmml">p</mi><mi id="S2.p1.3.m2.2.2.2.2.3" xref="S2.p1.3.m2.2.2.2.2.3.cmml">z</mi></msub><mo id="S2.p1.3.m2.2.2.2.5" stretchy="false" xref="S2.p1.3.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m2.2b"><interval closure="open" id="S2.p1.3.m2.2.2.3.cmml" xref="S2.p1.3.m2.2.2.2"><apply id="S2.p1.3.m2.1.1.1.1.cmml" xref="S2.p1.3.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.3.m2.1.1.1.1.1.cmml" xref="S2.p1.3.m2.1.1.1.1">subscript</csymbol><ci id="S2.p1.3.m2.1.1.1.1.2.cmml" xref="S2.p1.3.m2.1.1.1.1.2">𝑝</ci><ci id="S2.p1.3.m2.1.1.1.1.3.cmml" xref="S2.p1.3.m2.1.1.1.1.3">𝜌</ci></apply><apply id="S2.p1.3.m2.2.2.2.2.cmml" xref="S2.p1.3.m2.2.2.2.2"><csymbol cd="ambiguous" id="S2.p1.3.m2.2.2.2.2.1.cmml" xref="S2.p1.3.m2.2.2.2.2">subscript</csymbol><ci id="S2.p1.3.m2.2.2.2.2.2.cmml" xref="S2.p1.3.m2.2.2.2.2.2">𝑝</ci><ci id="S2.p1.3.m2.2.2.2.2.3.cmml" xref="S2.p1.3.m2.2.2.2.2.3">𝑧</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m2.2c">(p_{\rho},p_{z})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m2.2d">( italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT )</annotation></semantics></math> are their conjugate momenta. This Hamiltonian is expressed in a frame rotating about the <math alttext="z" class="ltx_Math" display="inline" id="S2.p1.4.m3.1"><semantics id="S2.p1.4.m3.1a"><mi id="S2.p1.4.m3.1.1" xref="S2.p1.4.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.p1.4.m3.1b"><ci id="S2.p1.4.m3.1.1.cmml" xref="S2.p1.4.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m3.1d">italic_z</annotation></semantics></math> axis with frequency <math alttext="\omega=B/2" class="ltx_Math" display="inline" id="S2.p1.5.m4.1"><semantics id="S2.p1.5.m4.1a"><mrow id="S2.p1.5.m4.1.1" xref="S2.p1.5.m4.1.1.cmml"><mi id="S2.p1.5.m4.1.1.2" xref="S2.p1.5.m4.1.1.2.cmml">ω</mi><mo id="S2.p1.5.m4.1.1.1" xref="S2.p1.5.m4.1.1.1.cmml">=</mo><mrow id="S2.p1.5.m4.1.1.3" xref="S2.p1.5.m4.1.1.3.cmml"><mi id="S2.p1.5.m4.1.1.3.2" xref="S2.p1.5.m4.1.1.3.2.cmml">B</mi><mo id="S2.p1.5.m4.1.1.3.1" xref="S2.p1.5.m4.1.1.3.1.cmml">/</mo><mn id="S2.p1.5.m4.1.1.3.3" xref="S2.p1.5.m4.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m4.1b"><apply id="S2.p1.5.m4.1.1.cmml" xref="S2.p1.5.m4.1.1"><eq id="S2.p1.5.m4.1.1.1.cmml" xref="S2.p1.5.m4.1.1.1"></eq><ci id="S2.p1.5.m4.1.1.2.cmml" xref="S2.p1.5.m4.1.1.2">𝜔</ci><apply id="S2.p1.5.m4.1.1.3.cmml" xref="S2.p1.5.m4.1.1.3"><divide id="S2.p1.5.m4.1.1.3.1.cmml" xref="S2.p1.5.m4.1.1.3.1"></divide><ci id="S2.p1.5.m4.1.1.3.2.cmml" xref="S2.p1.5.m4.1.1.3.2">𝐵</ci><cn id="S2.p1.5.m4.1.1.3.3.cmml" type="integer" xref="S2.p1.5.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m4.1c">\omega=B/2</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m4.1d">italic_ω = italic_B / 2</annotation></semantics></math> to eliminate the term linear in <math alttext="B" class="ltx_Math" display="inline" id="S2.p1.6.m5.1"><semantics id="S2.p1.6.m5.1a"><mi id="S2.p1.6.m5.1.1" xref="S2.p1.6.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m5.1b"><ci id="S2.p1.6.m5.1.1.cmml" xref="S2.p1.6.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m5.1d">italic_B</annotation></semantics></math>. Furthermore, we have assumed the angular momentum along the <math alttext="z" class="ltx_Math" display="inline" id="S2.p1.7.m6.1"><semantics id="S2.p1.7.m6.1a"><mi id="S2.p1.7.m6.1.1" xref="S2.p1.7.m6.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.p1.7.m6.1b"><ci id="S2.p1.7.m6.1.1.cmml" xref="S2.p1.7.m6.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m6.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m6.1d">italic_z</annotation></semantics></math> axis vanishes, i.e. <math alttext="L_{z}=0" class="ltx_Math" display="inline" id="S2.p1.8.m7.1"><semantics id="S2.p1.8.m7.1a"><mrow id="S2.p1.8.m7.1.1" xref="S2.p1.8.m7.1.1.cmml"><msub id="S2.p1.8.m7.1.1.2" xref="S2.p1.8.m7.1.1.2.cmml"><mi id="S2.p1.8.m7.1.1.2.2" xref="S2.p1.8.m7.1.1.2.2.cmml">L</mi><mi id="S2.p1.8.m7.1.1.2.3" xref="S2.p1.8.m7.1.1.2.3.cmml">z</mi></msub><mo id="S2.p1.8.m7.1.1.1" xref="S2.p1.8.m7.1.1.1.cmml">=</mo><mn id="S2.p1.8.m7.1.1.3" xref="S2.p1.8.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.8.m7.1b"><apply id="S2.p1.8.m7.1.1.cmml" xref="S2.p1.8.m7.1.1"><eq id="S2.p1.8.m7.1.1.1.cmml" xref="S2.p1.8.m7.1.1.1"></eq><apply id="S2.p1.8.m7.1.1.2.cmml" xref="S2.p1.8.m7.1.1.2"><csymbol cd="ambiguous" id="S2.p1.8.m7.1.1.2.1.cmml" xref="S2.p1.8.m7.1.1.2">subscript</csymbol><ci id="S2.p1.8.m7.1.1.2.2.cmml" xref="S2.p1.8.m7.1.1.2.2">𝐿</ci><ci id="S2.p1.8.m7.1.1.2.3.cmml" xref="S2.p1.8.m7.1.1.2.3">𝑧</ci></apply><cn id="S2.p1.8.m7.1.1.3.cmml" type="integer" xref="S2.p1.8.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m7.1c">L_{z}=0</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m7.1d">italic_L start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 0</annotation></semantics></math>. As is standard <cite class="ltx_cite ltx_citemacro_cite">Gao, Delos, and Baruch (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib21" title="">1992</a>); Gao and Delos (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib22" title="">1992</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib23" title="">1994</a>)</cite> the variables <math alttext="\left(\rho,z,p_{\rho},p_{z}\right)" class="ltx_Math" display="inline" id="S2.p1.9.m8.4"><semantics id="S2.p1.9.m8.4a"><mrow id="S2.p1.9.m8.4.4.2" xref="S2.p1.9.m8.4.4.3.cmml"><mo id="S2.p1.9.m8.4.4.2.3" xref="S2.p1.9.m8.4.4.3.cmml">(</mo><mi id="S2.p1.9.m8.1.1" xref="S2.p1.9.m8.1.1.cmml">ρ</mi><mo id="S2.p1.9.m8.4.4.2.4" xref="S2.p1.9.m8.4.4.3.cmml">,</mo><mi id="S2.p1.9.m8.2.2" xref="S2.p1.9.m8.2.2.cmml">z</mi><mo id="S2.p1.9.m8.4.4.2.5" xref="S2.p1.9.m8.4.4.3.cmml">,</mo><msub id="S2.p1.9.m8.3.3.1.1" xref="S2.p1.9.m8.3.3.1.1.cmml"><mi id="S2.p1.9.m8.3.3.1.1.2" xref="S2.p1.9.m8.3.3.1.1.2.cmml">p</mi><mi id="S2.p1.9.m8.3.3.1.1.3" xref="S2.p1.9.m8.3.3.1.1.3.cmml">ρ</mi></msub><mo id="S2.p1.9.m8.4.4.2.6" xref="S2.p1.9.m8.4.4.3.cmml">,</mo><msub id="S2.p1.9.m8.4.4.2.2" xref="S2.p1.9.m8.4.4.2.2.cmml"><mi id="S2.p1.9.m8.4.4.2.2.2" xref="S2.p1.9.m8.4.4.2.2.2.cmml">p</mi><mi id="S2.p1.9.m8.4.4.2.2.3" xref="S2.p1.9.m8.4.4.2.2.3.cmml">z</mi></msub><mo id="S2.p1.9.m8.4.4.2.7" xref="S2.p1.9.m8.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m8.4b"><vector id="S2.p1.9.m8.4.4.3.cmml" xref="S2.p1.9.m8.4.4.2"><ci id="S2.p1.9.m8.1.1.cmml" xref="S2.p1.9.m8.1.1">𝜌</ci><ci id="S2.p1.9.m8.2.2.cmml" xref="S2.p1.9.m8.2.2">𝑧</ci><apply id="S2.p1.9.m8.3.3.1.1.cmml" xref="S2.p1.9.m8.3.3.1.1"><csymbol cd="ambiguous" id="S2.p1.9.m8.3.3.1.1.1.cmml" xref="S2.p1.9.m8.3.3.1.1">subscript</csymbol><ci id="S2.p1.9.m8.3.3.1.1.2.cmml" xref="S2.p1.9.m8.3.3.1.1.2">𝑝</ci><ci id="S2.p1.9.m8.3.3.1.1.3.cmml" xref="S2.p1.9.m8.3.3.1.1.3">𝜌</ci></apply><apply id="S2.p1.9.m8.4.4.2.2.cmml" xref="S2.p1.9.m8.4.4.2.2"><csymbol cd="ambiguous" id="S2.p1.9.m8.4.4.2.2.1.cmml" xref="S2.p1.9.m8.4.4.2.2">subscript</csymbol><ci id="S2.p1.9.m8.4.4.2.2.2.cmml" xref="S2.p1.9.m8.4.4.2.2.2">𝑝</ci><ci id="S2.p1.9.m8.4.4.2.2.3.cmml" xref="S2.p1.9.m8.4.4.2.2.3">𝑧</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m8.4c">\left(\rho,z,p_{\rho},p_{z}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m8.4d">( italic_ρ , italic_z , italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT )</annotation></semantics></math> are scaled by the electric field <math alttext="\tilde{F}" class="ltx_Math" display="inline" id="S2.p1.10.m9.1"><semantics id="S2.p1.10.m9.1a"><mover accent="true" id="S2.p1.10.m9.1.1" xref="S2.p1.10.m9.1.1.cmml"><mi id="S2.p1.10.m9.1.1.2" xref="S2.p1.10.m9.1.1.2.cmml">F</mi><mo id="S2.p1.10.m9.1.1.1" xref="S2.p1.10.m9.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p1.10.m9.1b"><apply id="S2.p1.10.m9.1.1.cmml" xref="S2.p1.10.m9.1.1"><ci id="S2.p1.10.m9.1.1.1.cmml" xref="S2.p1.10.m9.1.1.1">~</ci><ci id="S2.p1.10.m9.1.1.2.cmml" xref="S2.p1.10.m9.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m9.1c">\tilde{F}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m9.1d">over~ start_ARG italic_F end_ARG</annotation></semantics></math> according to <math alttext="\left(\rho,z\right)=\left(\tilde{\rho}\tilde{F}^{\frac{1}{2}},\tilde{z}\tilde{% F}^{\frac{1}{2}}\right)" class="ltx_Math" display="inline" id="S2.p1.11.m10.4"><semantics id="S2.p1.11.m10.4a"><mrow id="S2.p1.11.m10.4.4" xref="S2.p1.11.m10.4.4.cmml"><mrow id="S2.p1.11.m10.4.4.4.2" xref="S2.p1.11.m10.4.4.4.1.cmml"><mo id="S2.p1.11.m10.4.4.4.2.1" xref="S2.p1.11.m10.4.4.4.1.cmml">(</mo><mi id="S2.p1.11.m10.1.1" 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xref="S2.p1.12.m11.3.3.3.1.1.3.3.2"></divide><cn id="S2.p1.12.m11.3.3.3.1.1.3.3.2.2.cmml" type="integer" xref="S2.p1.12.m11.3.3.3.1.1.3.3.2.2">1</cn><cn id="S2.p1.12.m11.3.3.3.1.1.3.3.2.3.cmml" type="integer" xref="S2.p1.12.m11.3.3.3.1.1.3.3.2.3">4</cn></apply></apply></apply></apply><apply id="S2.p1.12.m11.4.4.4.2.2.cmml" xref="S2.p1.12.m11.4.4.4.2.2"><times id="S2.p1.12.m11.4.4.4.2.2.1.cmml" xref="S2.p1.12.m11.4.4.4.2.2.1"></times><apply id="S2.p1.12.m11.4.4.4.2.2.2.cmml" xref="S2.p1.12.m11.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.p1.12.m11.4.4.4.2.2.2.1.cmml" xref="S2.p1.12.m11.4.4.4.2.2.2">subscript</csymbol><apply id="S2.p1.12.m11.4.4.4.2.2.2.2.cmml" xref="S2.p1.12.m11.4.4.4.2.2.2.2"><ci id="S2.p1.12.m11.4.4.4.2.2.2.2.1.cmml" xref="S2.p1.12.m11.4.4.4.2.2.2.2.1">~</ci><ci id="S2.p1.12.m11.4.4.4.2.2.2.2.2.cmml" xref="S2.p1.12.m11.4.4.4.2.2.2.2.2">𝑝</ci></apply><ci id="S2.p1.12.m11.4.4.4.2.2.2.3.cmml" xref="S2.p1.12.m11.4.4.4.2.2.2.3">𝑧</ci></apply><apply id="S2.p1.12.m11.4.4.4.2.2.3.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3"><csymbol cd="ambiguous" id="S2.p1.12.m11.4.4.4.2.2.3.1.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3">superscript</csymbol><apply id="S2.p1.12.m11.4.4.4.2.2.3.2.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3.2"><ci id="S2.p1.12.m11.4.4.4.2.2.3.2.1.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3.2.1">~</ci><ci id="S2.p1.12.m11.4.4.4.2.2.3.2.2.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3.2.2">𝐹</ci></apply><apply id="S2.p1.12.m11.4.4.4.2.2.3.3.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3.3"><minus id="S2.p1.12.m11.4.4.4.2.2.3.3.1.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3.3"></minus><apply id="S2.p1.12.m11.4.4.4.2.2.3.3.2.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3.3.2"><divide id="S2.p1.12.m11.4.4.4.2.2.3.3.2.1.cmml" xref="S2.p1.12.m11.4.4.4.2.2.3.3.2"></divide><cn id="S2.p1.12.m11.4.4.4.2.2.3.3.2.2.cmml" type="integer" xref="S2.p1.12.m11.4.4.4.2.2.3.3.2.2">1</cn><cn id="S2.p1.12.m11.4.4.4.2.2.3.3.2.3.cmml" type="integer" xref="S2.p1.12.m11.4.4.4.2.2.3.3.2.3">4</cn></apply></apply></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m11.4c">\left(p_{\rho},p_{z}\right)=\left(\tilde{p}_{\rho}\tilde{F}^{-\frac{1}{4}},% \tilde{p}_{z}\tilde{F}^{-\frac{1}{4}}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m11.4d">( italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = ( over~ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT over~ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG end_POSTSUPERSCRIPT , over~ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT over~ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math>. The magnetic field and electron energy are scaled like <math alttext="B=\tilde{B}\tilde{F}^{-\frac{3}{4}}" class="ltx_Math" display="inline" id="S2.p1.13.m12.1"><semantics id="S2.p1.13.m12.1a"><mrow id="S2.p1.13.m12.1.1" xref="S2.p1.13.m12.1.1.cmml"><mi id="S2.p1.13.m12.1.1.2" xref="S2.p1.13.m12.1.1.2.cmml">B</mi><mo id="S2.p1.13.m12.1.1.1" xref="S2.p1.13.m12.1.1.1.cmml">=</mo><mrow id="S2.p1.13.m12.1.1.3" xref="S2.p1.13.m12.1.1.3.cmml"><mover accent="true" id="S2.p1.13.m12.1.1.3.2" xref="S2.p1.13.m12.1.1.3.2.cmml"><mi id="S2.p1.13.m12.1.1.3.2.2" xref="S2.p1.13.m12.1.1.3.2.2.cmml">B</mi><mo id="S2.p1.13.m12.1.1.3.2.1" xref="S2.p1.13.m12.1.1.3.2.1.cmml">~</mo></mover><mo id="S2.p1.13.m12.1.1.3.1" xref="S2.p1.13.m12.1.1.3.1.cmml">⁢</mo><msup id="S2.p1.13.m12.1.1.3.3" xref="S2.p1.13.m12.1.1.3.3.cmml"><mover accent="true" id="S2.p1.13.m12.1.1.3.3.2" xref="S2.p1.13.m12.1.1.3.3.2.cmml"><mi id="S2.p1.13.m12.1.1.3.3.2.2" xref="S2.p1.13.m12.1.1.3.3.2.2.cmml">F</mi><mo id="S2.p1.13.m12.1.1.3.3.2.1" xref="S2.p1.13.m12.1.1.3.3.2.1.cmml">~</mo></mover><mrow id="S2.p1.13.m12.1.1.3.3.3" xref="S2.p1.13.m12.1.1.3.3.3.cmml"><mo id="S2.p1.13.m12.1.1.3.3.3a" xref="S2.p1.13.m12.1.1.3.3.3.cmml">−</mo><mfrac id="S2.p1.13.m12.1.1.3.3.3.2" xref="S2.p1.13.m12.1.1.3.3.3.2.cmml"><mn id="S2.p1.13.m12.1.1.3.3.3.2.2" xref="S2.p1.13.m12.1.1.3.3.3.2.2.cmml">3</mn><mn id="S2.p1.13.m12.1.1.3.3.3.2.3" xref="S2.p1.13.m12.1.1.3.3.3.2.3.cmml">4</mn></mfrac></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.13.m12.1b"><apply id="S2.p1.13.m12.1.1.cmml" xref="S2.p1.13.m12.1.1"><eq id="S2.p1.13.m12.1.1.1.cmml" xref="S2.p1.13.m12.1.1.1"></eq><ci id="S2.p1.13.m12.1.1.2.cmml" xref="S2.p1.13.m12.1.1.2">𝐵</ci><apply id="S2.p1.13.m12.1.1.3.cmml" xref="S2.p1.13.m12.1.1.3"><times id="S2.p1.13.m12.1.1.3.1.cmml" xref="S2.p1.13.m12.1.1.3.1"></times><apply id="S2.p1.13.m12.1.1.3.2.cmml" xref="S2.p1.13.m12.1.1.3.2"><ci id="S2.p1.13.m12.1.1.3.2.1.cmml" xref="S2.p1.13.m12.1.1.3.2.1">~</ci><ci id="S2.p1.13.m12.1.1.3.2.2.cmml" xref="S2.p1.13.m12.1.1.3.2.2">𝐵</ci></apply><apply id="S2.p1.13.m12.1.1.3.3.cmml" xref="S2.p1.13.m12.1.1.3.3"><csymbol cd="ambiguous" id="S2.p1.13.m12.1.1.3.3.1.cmml" xref="S2.p1.13.m12.1.1.3.3">superscript</csymbol><apply id="S2.p1.13.m12.1.1.3.3.2.cmml" xref="S2.p1.13.m12.1.1.3.3.2"><ci id="S2.p1.13.m12.1.1.3.3.2.1.cmml" xref="S2.p1.13.m12.1.1.3.3.2.1">~</ci><ci id="S2.p1.13.m12.1.1.3.3.2.2.cmml" xref="S2.p1.13.m12.1.1.3.3.2.2">𝐹</ci></apply><apply id="S2.p1.13.m12.1.1.3.3.3.cmml" xref="S2.p1.13.m12.1.1.3.3.3"><minus id="S2.p1.13.m12.1.1.3.3.3.1.cmml" xref="S2.p1.13.m12.1.1.3.3.3"></minus><apply id="S2.p1.13.m12.1.1.3.3.3.2.cmml" xref="S2.p1.13.m12.1.1.3.3.3.2"><divide id="S2.p1.13.m12.1.1.3.3.3.2.1.cmml" xref="S2.p1.13.m12.1.1.3.3.3.2"></divide><cn id="S2.p1.13.m12.1.1.3.3.3.2.2.cmml" type="integer" xref="S2.p1.13.m12.1.1.3.3.3.2.2">3</cn><cn id="S2.p1.13.m12.1.1.3.3.3.2.3.cmml" type="integer" xref="S2.p1.13.m12.1.1.3.3.3.2.3">4</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m12.1c">B=\tilde{B}\tilde{F}^{-\frac{3}{4}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m12.1d">italic_B = over~ start_ARG italic_B end_ARG over~ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT - divide start_ARG 3 end_ARG start_ARG 4 end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="E=\tilde{E}\tilde{F}^{-\frac{1}{2}}" class="ltx_Math" display="inline" id="S2.p1.14.m13.1"><semantics id="S2.p1.14.m13.1a"><mrow id="S2.p1.14.m13.1.1" xref="S2.p1.14.m13.1.1.cmml"><mi id="S2.p1.14.m13.1.1.2" xref="S2.p1.14.m13.1.1.2.cmml">E</mi><mo id="S2.p1.14.m13.1.1.1" xref="S2.p1.14.m13.1.1.1.cmml">=</mo><mrow id="S2.p1.14.m13.1.1.3" xref="S2.p1.14.m13.1.1.3.cmml"><mover accent="true" id="S2.p1.14.m13.1.1.3.2" xref="S2.p1.14.m13.1.1.3.2.cmml"><mi id="S2.p1.14.m13.1.1.3.2.2" xref="S2.p1.14.m13.1.1.3.2.2.cmml">E</mi><mo id="S2.p1.14.m13.1.1.3.2.1" xref="S2.p1.14.m13.1.1.3.2.1.cmml">~</mo></mover><mo id="S2.p1.14.m13.1.1.3.1" xref="S2.p1.14.m13.1.1.3.1.cmml">⁢</mo><msup id="S2.p1.14.m13.1.1.3.3" xref="S2.p1.14.m13.1.1.3.3.cmml"><mover accent="true" id="S2.p1.14.m13.1.1.3.3.2" xref="S2.p1.14.m13.1.1.3.3.2.cmml"><mi id="S2.p1.14.m13.1.1.3.3.2.2" xref="S2.p1.14.m13.1.1.3.3.2.2.cmml">F</mi><mo id="S2.p1.14.m13.1.1.3.3.2.1" xref="S2.p1.14.m13.1.1.3.3.2.1.cmml">~</mo></mover><mrow id="S2.p1.14.m13.1.1.3.3.3" xref="S2.p1.14.m13.1.1.3.3.3.cmml"><mo id="S2.p1.14.m13.1.1.3.3.3a" xref="S2.p1.14.m13.1.1.3.3.3.cmml">−</mo><mfrac id="S2.p1.14.m13.1.1.3.3.3.2" xref="S2.p1.14.m13.1.1.3.3.3.2.cmml"><mn id="S2.p1.14.m13.1.1.3.3.3.2.2" xref="S2.p1.14.m13.1.1.3.3.3.2.2.cmml">1</mn><mn id="S2.p1.14.m13.1.1.3.3.3.2.3" xref="S2.p1.14.m13.1.1.3.3.3.2.3.cmml">2</mn></mfrac></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.14.m13.1b"><apply id="S2.p1.14.m13.1.1.cmml" xref="S2.p1.14.m13.1.1"><eq id="S2.p1.14.m13.1.1.1.cmml" xref="S2.p1.14.m13.1.1.1"></eq><ci id="S2.p1.14.m13.1.1.2.cmml" xref="S2.p1.14.m13.1.1.2">𝐸</ci><apply id="S2.p1.14.m13.1.1.3.cmml" xref="S2.p1.14.m13.1.1.3"><times id="S2.p1.14.m13.1.1.3.1.cmml" xref="S2.p1.14.m13.1.1.3.1"></times><apply id="S2.p1.14.m13.1.1.3.2.cmml" xref="S2.p1.14.m13.1.1.3.2"><ci id="S2.p1.14.m13.1.1.3.2.1.cmml" xref="S2.p1.14.m13.1.1.3.2.1">~</ci><ci id="S2.p1.14.m13.1.1.3.2.2.cmml" xref="S2.p1.14.m13.1.1.3.2.2">𝐸</ci></apply><apply id="S2.p1.14.m13.1.1.3.3.cmml" xref="S2.p1.14.m13.1.1.3.3"><csymbol cd="ambiguous" id="S2.p1.14.m13.1.1.3.3.1.cmml" xref="S2.p1.14.m13.1.1.3.3">superscript</csymbol><apply id="S2.p1.14.m13.1.1.3.3.2.cmml" xref="S2.p1.14.m13.1.1.3.3.2"><ci id="S2.p1.14.m13.1.1.3.3.2.1.cmml" xref="S2.p1.14.m13.1.1.3.3.2.1">~</ci><ci id="S2.p1.14.m13.1.1.3.3.2.2.cmml" xref="S2.p1.14.m13.1.1.3.3.2.2">𝐹</ci></apply><apply id="S2.p1.14.m13.1.1.3.3.3.cmml" xref="S2.p1.14.m13.1.1.3.3.3"><minus id="S2.p1.14.m13.1.1.3.3.3.1.cmml" xref="S2.p1.14.m13.1.1.3.3.3"></minus><apply id="S2.p1.14.m13.1.1.3.3.3.2.cmml" xref="S2.p1.14.m13.1.1.3.3.3.2"><divide id="S2.p1.14.m13.1.1.3.3.3.2.1.cmml" xref="S2.p1.14.m13.1.1.3.3.3.2"></divide><cn id="S2.p1.14.m13.1.1.3.3.3.2.2.cmml" type="integer" xref="S2.p1.14.m13.1.1.3.3.3.2.2">1</cn><cn id="S2.p1.14.m13.1.1.3.3.3.2.3.cmml" type="integer" xref="S2.p1.14.m13.1.1.3.3.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m13.1c">E=\tilde{E}\tilde{F}^{-\frac{1}{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m13.1d">italic_E = over~ start_ARG italic_E end_ARG over~ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math>. Here, the tilded symbols represent the physical, unscaled variables and the regular untilded symbols represent the scaled variables. <cite class="ltx_cite ltx_citemacro_cite">Mitchell <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib24" title="">2004</a>)</cite></p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">As is common in the literature <cite class="ltx_cite ltx_citemacro_cite">Haggerty and Delos (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib25" title="">2000</a>)</cite>, we next transform into parabolic coordinates <math alttext="(u,v)" class="ltx_Math" display="inline" id="S2.p2.1.m1.2"><semantics id="S2.p2.1.m1.2a"><mrow id="S2.p2.1.m1.2.3.2" xref="S2.p2.1.m1.2.3.1.cmml"><mo id="S2.p2.1.m1.2.3.2.1" stretchy="false" xref="S2.p2.1.m1.2.3.1.cmml">(</mo><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">u</mi><mo id="S2.p2.1.m1.2.3.2.2" xref="S2.p2.1.m1.2.3.1.cmml">,</mo><mi id="S2.p2.1.m1.2.2" xref="S2.p2.1.m1.2.2.cmml">v</mi><mo id="S2.p2.1.m1.2.3.2.3" stretchy="false" xref="S2.p2.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.2b"><interval closure="open" id="S2.p2.1.m1.2.3.1.cmml" xref="S2.p2.1.m1.2.3.2"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝑢</ci><ci id="S2.p2.1.m1.2.2.cmml" xref="S2.p2.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.2d">( italic_u , italic_v )</annotation></semantics></math> defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <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="u=\pm\sqrt{\rho+z}\,,\quad v=\pm\sqrt{\rho-z}," class="ltx_Math" display="block" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><mrow id="S2.E2.m1.1.1.1"><mrow id="S2.E2.m1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.3.cmml"><mrow id="S2.E2.m1.1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.E2.m1.1.1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E2.m1.1.1.1.1.1.1.3" xref="S2.E2.m1.1.1.1.1.1.1.3.cmml"><mo id="S2.E2.m1.1.1.1.1.1.1.3a" xref="S2.E2.m1.1.1.1.1.1.1.3.cmml">±</mo><msqrt id="S2.E2.m1.1.1.1.1.1.1.3.2" xref="S2.E2.m1.1.1.1.1.1.1.3.2.cmml"><mrow id="S2.E2.m1.1.1.1.1.1.1.3.2.2" xref="S2.E2.m1.1.1.1.1.1.1.3.2.2.cmml"><mi id="S2.E2.m1.1.1.1.1.1.1.3.2.2.2" xref="S2.E2.m1.1.1.1.1.1.1.3.2.2.2.cmml">ρ</mi><mo id="S2.E2.m1.1.1.1.1.1.1.3.2.2.1" xref="S2.E2.m1.1.1.1.1.1.1.3.2.2.1.cmml">+</mo><mi id="S2.E2.m1.1.1.1.1.1.1.3.2.2.3" xref="S2.E2.m1.1.1.1.1.1.1.3.2.2.3.cmml">z</mi></mrow></msqrt></mrow></mrow><mo id="S2.E2.m1.1.1.1.1.2.3" lspace="0.170em" rspace="1.167em" xref="S2.E2.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S2.E2.m1.1.1.1.1.2.2" xref="S2.E2.m1.1.1.1.1.2.2.cmml"><mi id="S2.E2.m1.1.1.1.1.2.2.2" xref="S2.E2.m1.1.1.1.1.2.2.2.cmml">v</mi><mo id="S2.E2.m1.1.1.1.1.2.2.1" xref="S2.E2.m1.1.1.1.1.2.2.1.cmml">=</mo><mrow id="S2.E2.m1.1.1.1.1.2.2.3" xref="S2.E2.m1.1.1.1.1.2.2.3.cmml"><mo id="S2.E2.m1.1.1.1.1.2.2.3a" xref="S2.E2.m1.1.1.1.1.2.2.3.cmml">±</mo><msqrt id="S2.E2.m1.1.1.1.1.2.2.3.2" xref="S2.E2.m1.1.1.1.1.2.2.3.2.cmml"><mrow id="S2.E2.m1.1.1.1.1.2.2.3.2.2" xref="S2.E2.m1.1.1.1.1.2.2.3.2.2.cmml"><mi id="S2.E2.m1.1.1.1.1.2.2.3.2.2.2" xref="S2.E2.m1.1.1.1.1.2.2.3.2.2.2.cmml">ρ</mi><mo id="S2.E2.m1.1.1.1.1.2.2.3.2.2.1" xref="S2.E2.m1.1.1.1.1.2.2.3.2.2.1.cmml">−</mo><mi id="S2.E2.m1.1.1.1.1.2.2.3.2.2.3" xref="S2.E2.m1.1.1.1.1.2.2.3.2.2.3.cmml">z</mi></mrow></msqrt></mrow></mrow></mrow><mo id="S2.E2.m1.1.1.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m1.1b"><apply id="S2.E2.m1.1.1.1.1.3.cmml" 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xref="S2.E2.m1.1.1.1.1.2.2.3.2.2.3">𝑧</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.1c">u=\pm\sqrt{\rho+z}\,,\quad v=\pm\sqrt{\rho-z},</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">italic_u = ± square-root start_ARG italic_ρ + italic_z end_ARG , italic_v = ± square-root start_ARG italic_ρ - italic_z 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">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p2.13">with conjugate momenta</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="p_{u}=vp_{\rho}+up_{z}\,,\quad 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id="S2.E3.m1.1.1.1.1.2.2.3.3.2.cmml" xref="S2.E3.m1.1.1.1.1.2.2.3.3.2">𝑣</ci><apply id="S2.E3.m1.1.1.1.1.2.2.3.3.3.cmml" xref="S2.E3.m1.1.1.1.1.2.2.3.3.3"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.2.2.3.3.3.1.cmml" xref="S2.E3.m1.1.1.1.1.2.2.3.3.3">subscript</csymbol><ci id="S2.E3.m1.1.1.1.1.2.2.3.3.3.2.cmml" xref="S2.E3.m1.1.1.1.1.2.2.3.3.3.2">𝑝</ci><ci id="S2.E3.m1.1.1.1.1.2.2.3.3.3.3.cmml" xref="S2.E3.m1.1.1.1.1.2.2.3.3.3.3">𝑧</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">p_{u}=vp_{\rho}+up_{z}\,,\quad p_{v}=up_{\rho}-vp_{z}.</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">italic_p start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT = italic_v italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT + italic_u italic_p start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = italic_u italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT - italic_v italic_p start_POSTSUBSCRIPT italic_z 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p2.3">Finally, the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S2.E1" title="In II Hydrogen in Parallel Electric and Magnetic Fields ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">1</span></a>) is transformed into <math alttext="h=2r\left(H-E\right)" class="ltx_Math" display="inline" id="S2.p2.2.m1.1"><semantics id="S2.p2.2.m1.1a"><mrow id="S2.p2.2.m1.1.1" xref="S2.p2.2.m1.1.1.cmml"><mi id="S2.p2.2.m1.1.1.3" xref="S2.p2.2.m1.1.1.3.cmml">h</mi><mo id="S2.p2.2.m1.1.1.2" xref="S2.p2.2.m1.1.1.2.cmml">=</mo><mrow id="S2.p2.2.m1.1.1.1" xref="S2.p2.2.m1.1.1.1.cmml"><mn id="S2.p2.2.m1.1.1.1.3" xref="S2.p2.2.m1.1.1.1.3.cmml">2</mn><mo id="S2.p2.2.m1.1.1.1.2" xref="S2.p2.2.m1.1.1.1.2.cmml">⁢</mo><mi id="S2.p2.2.m1.1.1.1.4" xref="S2.p2.2.m1.1.1.1.4.cmml">r</mi><mo id="S2.p2.2.m1.1.1.1.2a" xref="S2.p2.2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.p2.2.m1.1.1.1.1.1" xref="S2.p2.2.m1.1.1.1.1.1.1.cmml"><mo id="S2.p2.2.m1.1.1.1.1.1.2" xref="S2.p2.2.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.p2.2.m1.1.1.1.1.1.1" xref="S2.p2.2.m1.1.1.1.1.1.1.cmml"><mi id="S2.p2.2.m1.1.1.1.1.1.1.2" xref="S2.p2.2.m1.1.1.1.1.1.1.2.cmml">H</mi><mo id="S2.p2.2.m1.1.1.1.1.1.1.1" xref="S2.p2.2.m1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S2.p2.2.m1.1.1.1.1.1.1.3" xref="S2.p2.2.m1.1.1.1.1.1.1.3.cmml">E</mi></mrow><mo id="S2.p2.2.m1.1.1.1.1.1.3" xref="S2.p2.2.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.2.m1.1b"><apply id="S2.p2.2.m1.1.1.cmml" xref="S2.p2.2.m1.1.1"><eq id="S2.p2.2.m1.1.1.2.cmml" xref="S2.p2.2.m1.1.1.2"></eq><ci id="S2.p2.2.m1.1.1.3.cmml" xref="S2.p2.2.m1.1.1.3">ℎ</ci><apply id="S2.p2.2.m1.1.1.1.cmml" xref="S2.p2.2.m1.1.1.1"><times id="S2.p2.2.m1.1.1.1.2.cmml" xref="S2.p2.2.m1.1.1.1.2"></times><cn id="S2.p2.2.m1.1.1.1.3.cmml" type="integer" xref="S2.p2.2.m1.1.1.1.3">2</cn><ci id="S2.p2.2.m1.1.1.1.4.cmml" xref="S2.p2.2.m1.1.1.1.4">𝑟</ci><apply id="S2.p2.2.m1.1.1.1.1.1.1.cmml" xref="S2.p2.2.m1.1.1.1.1.1"><minus id="S2.p2.2.m1.1.1.1.1.1.1.1.cmml" xref="S2.p2.2.m1.1.1.1.1.1.1.1"></minus><ci id="S2.p2.2.m1.1.1.1.1.1.1.2.cmml" xref="S2.p2.2.m1.1.1.1.1.1.1.2">𝐻</ci><ci id="S2.p2.2.m1.1.1.1.1.1.1.3.cmml" xref="S2.p2.2.m1.1.1.1.1.1.1.3">𝐸</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m1.1c">h=2r\left(H-E\right)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m1.1d">italic_h = 2 italic_r ( italic_H - italic_E )</annotation></semantics></math>, where <math alttext="r=\sqrt{\rho^{2}+z^{2}}" class="ltx_Math" display="inline" id="S2.p2.3.m2.1"><semantics id="S2.p2.3.m2.1a"><mrow id="S2.p2.3.m2.1.1" xref="S2.p2.3.m2.1.1.cmml"><mi id="S2.p2.3.m2.1.1.2" xref="S2.p2.3.m2.1.1.2.cmml">r</mi><mo id="S2.p2.3.m2.1.1.1" xref="S2.p2.3.m2.1.1.1.cmml">=</mo><msqrt id="S2.p2.3.m2.1.1.3" xref="S2.p2.3.m2.1.1.3.cmml"><mrow id="S2.p2.3.m2.1.1.3.2" xref="S2.p2.3.m2.1.1.3.2.cmml"><msup id="S2.p2.3.m2.1.1.3.2.2" xref="S2.p2.3.m2.1.1.3.2.2.cmml"><mi id="S2.p2.3.m2.1.1.3.2.2.2" xref="S2.p2.3.m2.1.1.3.2.2.2.cmml">ρ</mi><mn id="S2.p2.3.m2.1.1.3.2.2.3" xref="S2.p2.3.m2.1.1.3.2.2.3.cmml">2</mn></msup><mo id="S2.p2.3.m2.1.1.3.2.1" xref="S2.p2.3.m2.1.1.3.2.1.cmml">+</mo><msup id="S2.p2.3.m2.1.1.3.2.3" xref="S2.p2.3.m2.1.1.3.2.3.cmml"><mi id="S2.p2.3.m2.1.1.3.2.3.2" xref="S2.p2.3.m2.1.1.3.2.3.2.cmml">z</mi><mn id="S2.p2.3.m2.1.1.3.2.3.3" xref="S2.p2.3.m2.1.1.3.2.3.3.cmml">2</mn></msup></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.3.m2.1b"><apply id="S2.p2.3.m2.1.1.cmml" 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xref="S2.E4.m1.4.4">𝑣</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.7c">h(u,v,p_{u},p_{v})=\frac{1}{2}(p^{2}_{u}+p^{2}_{v})+V\left(u,v\right)</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.7d">italic_h ( italic_u , italic_v , italic_p start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT + italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) + italic_V ( italic_u , italic_v )</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 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id="S2.E5.m1.3c">V\left(u,v\right)=-E(u^{2}+v^{2})+\frac{1}{8}B^{2}(u^{2}v^{4}+u^{4}v^{2})+% \frac{1}{2}(u^{4}-v^{4})-2.</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.3d">italic_V ( italic_u , italic_v ) = - italic_E ( italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 8 end_ARG italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + italic_u start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_u start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - italic_v start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) - 2 .</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.p2.12">We recover <math alttext="H=E" class="ltx_Math" display="inline" id="S2.p2.4.m1.1"><semantics id="S2.p2.4.m1.1a"><mrow id="S2.p2.4.m1.1.1" xref="S2.p2.4.m1.1.1.cmml"><mi id="S2.p2.4.m1.1.1.2" xref="S2.p2.4.m1.1.1.2.cmml">H</mi><mo id="S2.p2.4.m1.1.1.1" xref="S2.p2.4.m1.1.1.1.cmml">=</mo><mi id="S2.p2.4.m1.1.1.3" xref="S2.p2.4.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.4.m1.1b"><apply id="S2.p2.4.m1.1.1.cmml" xref="S2.p2.4.m1.1.1"><eq id="S2.p2.4.m1.1.1.1.cmml" xref="S2.p2.4.m1.1.1.1"></eq><ci id="S2.p2.4.m1.1.1.2.cmml" xref="S2.p2.4.m1.1.1.2">𝐻</ci><ci id="S2.p2.4.m1.1.1.3.cmml" xref="S2.p2.4.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m1.1c">H=E</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m1.1d">italic_H = italic_E</annotation></semantics></math> by requiring <math alttext="h=0" class="ltx_Math" display="inline" id="S2.p2.5.m2.1"><semantics id="S2.p2.5.m2.1a"><mrow id="S2.p2.5.m2.1.1" xref="S2.p2.5.m2.1.1.cmml"><mi id="S2.p2.5.m2.1.1.2" xref="S2.p2.5.m2.1.1.2.cmml">h</mi><mo id="S2.p2.5.m2.1.1.1" xref="S2.p2.5.m2.1.1.1.cmml">=</mo><mn id="S2.p2.5.m2.1.1.3" xref="S2.p2.5.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.5.m2.1b"><apply id="S2.p2.5.m2.1.1.cmml" xref="S2.p2.5.m2.1.1"><eq id="S2.p2.5.m2.1.1.1.cmml" xref="S2.p2.5.m2.1.1.1"></eq><ci id="S2.p2.5.m2.1.1.2.cmml" xref="S2.p2.5.m2.1.1.2">ℎ</ci><cn id="S2.p2.5.m2.1.1.3.cmml" type="integer" xref="S2.p2.5.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.5.m2.1c">h=0</annotation><annotation encoding="application/x-llamapun" id="S2.p2.5.m2.1d">italic_h = 0</annotation></semantics></math>. The electron energy <math alttext="E" class="ltx_Math" display="inline" id="S2.p2.6.m3.1"><semantics id="S2.p2.6.m3.1a"><mi id="S2.p2.6.m3.1.1" xref="S2.p2.6.m3.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S2.p2.6.m3.1b"><ci id="S2.p2.6.m3.1.1.cmml" xref="S2.p2.6.m3.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.6.m3.1c">E</annotation><annotation encoding="application/x-llamapun" id="S2.p2.6.m3.1d">italic_E</annotation></semantics></math> now behaves as a parameter of our new Hamiltonian <math alttext="h" class="ltx_Math" display="inline" id="S2.p2.7.m4.1"><semantics id="S2.p2.7.m4.1a"><mi id="S2.p2.7.m4.1.1" xref="S2.p2.7.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.p2.7.m4.1b"><ci id="S2.p2.7.m4.1.1.cmml" xref="S2.p2.7.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.7.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.p2.7.m4.1d">italic_h</annotation></semantics></math>. The other parameter <math alttext="B" class="ltx_Math" display="inline" id="S2.p2.8.m5.1"><semantics id="S2.p2.8.m5.1a"><mi id="S2.p2.8.m5.1.1" xref="S2.p2.8.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.p2.8.m5.1b"><ci id="S2.p2.8.m5.1.1.cmml" xref="S2.p2.8.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.8.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.p2.8.m5.1d">italic_B</annotation></semantics></math> acts as a coupling constant between the spatial coordinates, which generates the chaotic mixing we are interested in studying <cite class="ltx_cite ltx_citemacro_cite">Mitchell <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib24" title="">2004</a>)</cite>. Note that the new Hamiltonian <math alttext="h" class="ltx_Math" display="inline" id="S2.p2.9.m6.1"><semantics id="S2.p2.9.m6.1a"><mi id="S2.p2.9.m6.1.1" xref="S2.p2.9.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.p2.9.m6.1b"><ci id="S2.p2.9.m6.1.1.cmml" xref="S2.p2.9.m6.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.9.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.p2.9.m6.1d">italic_h</annotation></semantics></math> has a conjugate time variable <math alttext="s" class="ltx_Math" display="inline" id="S2.p2.10.m7.1"><semantics id="S2.p2.10.m7.1a"><mi id="S2.p2.10.m7.1.1" xref="S2.p2.10.m7.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.p2.10.m7.1b"><ci id="S2.p2.10.m7.1.1.cmml" xref="S2.p2.10.m7.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.10.m7.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.p2.10.m7.1d">italic_s</annotation></semantics></math> defined by <math alttext="ds/dt=1/(2r)" class="ltx_Math" display="inline" id="S2.p2.11.m8.1"><semantics id="S2.p2.11.m8.1a"><mrow id="S2.p2.11.m8.1.1" xref="S2.p2.11.m8.1.1.cmml"><mrow id="S2.p2.11.m8.1.1.3" xref="S2.p2.11.m8.1.1.3.cmml"><mrow id="S2.p2.11.m8.1.1.3.2" xref="S2.p2.11.m8.1.1.3.2.cmml"><mrow id="S2.p2.11.m8.1.1.3.2.2" xref="S2.p2.11.m8.1.1.3.2.2.cmml"><mi id="S2.p2.11.m8.1.1.3.2.2.2" xref="S2.p2.11.m8.1.1.3.2.2.2.cmml">d</mi><mo id="S2.p2.11.m8.1.1.3.2.2.1" xref="S2.p2.11.m8.1.1.3.2.2.1.cmml">⁢</mo><mi id="S2.p2.11.m8.1.1.3.2.2.3" xref="S2.p2.11.m8.1.1.3.2.2.3.cmml">s</mi></mrow><mo id="S2.p2.11.m8.1.1.3.2.1" xref="S2.p2.11.m8.1.1.3.2.1.cmml">/</mo><mi id="S2.p2.11.m8.1.1.3.2.3" xref="S2.p2.11.m8.1.1.3.2.3.cmml">d</mi></mrow><mo id="S2.p2.11.m8.1.1.3.1" xref="S2.p2.11.m8.1.1.3.1.cmml">⁢</mo><mi id="S2.p2.11.m8.1.1.3.3" xref="S2.p2.11.m8.1.1.3.3.cmml">t</mi></mrow><mo id="S2.p2.11.m8.1.1.2" xref="S2.p2.11.m8.1.1.2.cmml">=</mo><mrow id="S2.p2.11.m8.1.1.1" xref="S2.p2.11.m8.1.1.1.cmml"><mn id="S2.p2.11.m8.1.1.1.3" xref="S2.p2.11.m8.1.1.1.3.cmml">1</mn><mo id="S2.p2.11.m8.1.1.1.2" xref="S2.p2.11.m8.1.1.1.2.cmml">/</mo><mrow id="S2.p2.11.m8.1.1.1.1.1" xref="S2.p2.11.m8.1.1.1.1.1.1.cmml"><mo id="S2.p2.11.m8.1.1.1.1.1.2" stretchy="false" xref="S2.p2.11.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.p2.11.m8.1.1.1.1.1.1" xref="S2.p2.11.m8.1.1.1.1.1.1.cmml"><mn id="S2.p2.11.m8.1.1.1.1.1.1.2" xref="S2.p2.11.m8.1.1.1.1.1.1.2.cmml">2</mn><mo id="S2.p2.11.m8.1.1.1.1.1.1.1" xref="S2.p2.11.m8.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.p2.11.m8.1.1.1.1.1.1.3" xref="S2.p2.11.m8.1.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S2.p2.11.m8.1.1.1.1.1.3" stretchy="false" xref="S2.p2.11.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.11.m8.1b"><apply id="S2.p2.11.m8.1.1.cmml" xref="S2.p2.11.m8.1.1"><eq id="S2.p2.11.m8.1.1.2.cmml" xref="S2.p2.11.m8.1.1.2"></eq><apply id="S2.p2.11.m8.1.1.3.cmml" xref="S2.p2.11.m8.1.1.3"><times id="S2.p2.11.m8.1.1.3.1.cmml" xref="S2.p2.11.m8.1.1.3.1"></times><apply id="S2.p2.11.m8.1.1.3.2.cmml" xref="S2.p2.11.m8.1.1.3.2"><divide id="S2.p2.11.m8.1.1.3.2.1.cmml" xref="S2.p2.11.m8.1.1.3.2.1"></divide><apply id="S2.p2.11.m8.1.1.3.2.2.cmml" xref="S2.p2.11.m8.1.1.3.2.2"><times id="S2.p2.11.m8.1.1.3.2.2.1.cmml" xref="S2.p2.11.m8.1.1.3.2.2.1"></times><ci id="S2.p2.11.m8.1.1.3.2.2.2.cmml" xref="S2.p2.11.m8.1.1.3.2.2.2">𝑑</ci><ci id="S2.p2.11.m8.1.1.3.2.2.3.cmml" xref="S2.p2.11.m8.1.1.3.2.2.3">𝑠</ci></apply><ci id="S2.p2.11.m8.1.1.3.2.3.cmml" xref="S2.p2.11.m8.1.1.3.2.3">𝑑</ci></apply><ci id="S2.p2.11.m8.1.1.3.3.cmml" xref="S2.p2.11.m8.1.1.3.3">𝑡</ci></apply><apply id="S2.p2.11.m8.1.1.1.cmml" xref="S2.p2.11.m8.1.1.1"><divide id="S2.p2.11.m8.1.1.1.2.cmml" xref="S2.p2.11.m8.1.1.1.2"></divide><cn id="S2.p2.11.m8.1.1.1.3.cmml" type="integer" xref="S2.p2.11.m8.1.1.1.3">1</cn><apply id="S2.p2.11.m8.1.1.1.1.1.1.cmml" xref="S2.p2.11.m8.1.1.1.1.1"><times id="S2.p2.11.m8.1.1.1.1.1.1.1.cmml" xref="S2.p2.11.m8.1.1.1.1.1.1.1"></times><cn id="S2.p2.11.m8.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.p2.11.m8.1.1.1.1.1.1.2">2</cn><ci id="S2.p2.11.m8.1.1.1.1.1.1.3.cmml" xref="S2.p2.11.m8.1.1.1.1.1.1.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.11.m8.1c">ds/dt=1/(2r)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.11.m8.1d">italic_d italic_s / italic_d italic_t = 1 / ( 2 italic_r )</annotation></semantics></math>. In the following, all references to the continuous time variable will refer to the new time <math alttext="s" class="ltx_Math" display="inline" id="S2.p2.12.m9.1"><semantics id="S2.p2.12.m9.1a"><mi id="S2.p2.12.m9.1.1" xref="S2.p2.12.m9.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.p2.12.m9.1b"><ci id="S2.p2.12.m9.1.1.cmml" xref="S2.p2.12.m9.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.12.m9.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.p2.12.m9.1d">italic_s</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">III </span>Escape Rate from Classical Trajectory Monte Carlo</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.11">We conduct classical trajectory Monte Carlo simulations to compute the classical escape rate <math alttext="\gamma" 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">γ</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">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_γ</annotation></semantics></math> at given values of <math alttext="B" 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">B</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">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_B</annotation></semantics></math> and <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>. We use an initial ensemble of <math alttext="10^{7}" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><msup id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mn id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml">10</mn><mn id="S3.p1.4.m4.1.1.3" xref="S3.p1.4.m4.1.1.3.cmml">7</mn></msup><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"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.1.cmml" xref="S3.p1.4.m4.1.1">superscript</csymbol><cn id="S3.p1.4.m4.1.1.2.cmml" type="integer" xref="S3.p1.4.m4.1.1.2">10</cn><cn id="S3.p1.4.m4.1.1.3.cmml" type="integer" xref="S3.p1.4.m4.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">10^{7}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT</annotation></semantics></math> trajectories radially propagating from the origin; this constitutes a classical model of the quantum electron state immediately after it absorbs a short laser pulse <cite class="ltx_cite ltx_citemacro_cite">Mitchell <em class="ltx_emph ltx_font_italic">et al.</em> (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib24" title="">2004</a>)</cite>. Ionizing electron trajectories escape along the negative <math alttext="z" 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">z</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">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">italic_z</annotation></semantics></math> direction, and we thus define escape, i.e. ionization, to be when a trajectory reaches <math alttext="z=-1" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><mrow id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml"><mi id="S3.p1.6.m6.1.1.2" xref="S3.p1.6.m6.1.1.2.cmml">z</mi><mo id="S3.p1.6.m6.1.1.1" xref="S3.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S3.p1.6.m6.1.1.3" xref="S3.p1.6.m6.1.1.3.cmml"><mo id="S3.p1.6.m6.1.1.3a" xref="S3.p1.6.m6.1.1.3.cmml">−</mo><mn id="S3.p1.6.m6.1.1.3.2" xref="S3.p1.6.m6.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><apply id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1"><eq id="S3.p1.6.m6.1.1.1.cmml" xref="S3.p1.6.m6.1.1.1"></eq><ci id="S3.p1.6.m6.1.1.2.cmml" xref="S3.p1.6.m6.1.1.2">𝑧</ci><apply id="S3.p1.6.m6.1.1.3.cmml" xref="S3.p1.6.m6.1.1.3"><minus id="S3.p1.6.m6.1.1.3.1.cmml" xref="S3.p1.6.m6.1.1.3"></minus><cn id="S3.p1.6.m6.1.1.3.2.cmml" type="integer" xref="S3.p1.6.m6.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">z=-1</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">italic_z = - 1</annotation></semantics></math>. Once trajectories cross this escape boundary, they will continue to infinity. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S3.F1" title="Figure 1 ‣ III Escape Rate from Classical Trajectory Monte Carlo ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">1</span></a>a plots the number of surviving (unionized) trajectories as a function of the time <math alttext="s" class="ltx_Math" display="inline" id="S3.p1.7.m7.1"><semantics id="S3.p1.7.m7.1a"><mi id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.1b"><ci id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.1c">s</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.1d">italic_s</annotation></semantics></math> computed at <math alttext="E=1.0" class="ltx_Math" display="inline" id="S3.p1.8.m8.1"><semantics id="S3.p1.8.m8.1a"><mrow id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml"><mi id="S3.p1.8.m8.1.1.2" xref="S3.p1.8.m8.1.1.2.cmml">E</mi><mo id="S3.p1.8.m8.1.1.1" xref="S3.p1.8.m8.1.1.1.cmml">=</mo><mn id="S3.p1.8.m8.1.1.3" xref="S3.p1.8.m8.1.1.3.cmml">1.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.1b"><apply id="S3.p1.8.m8.1.1.cmml" xref="S3.p1.8.m8.1.1"><eq id="S3.p1.8.m8.1.1.1.cmml" xref="S3.p1.8.m8.1.1.1"></eq><ci id="S3.p1.8.m8.1.1.2.cmml" xref="S3.p1.8.m8.1.1.2">𝐸</ci><cn id="S3.p1.8.m8.1.1.3.cmml" type="float" xref="S3.p1.8.m8.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m8.1c">E=1.0</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m8.1d">italic_E = 1.0</annotation></semantics></math> and <math alttext="B=3.5" class="ltx_Math" display="inline" id="S3.p1.9.m9.1"><semantics id="S3.p1.9.m9.1a"><mrow id="S3.p1.9.m9.1.1" xref="S3.p1.9.m9.1.1.cmml"><mi id="S3.p1.9.m9.1.1.2" xref="S3.p1.9.m9.1.1.2.cmml">B</mi><mo id="S3.p1.9.m9.1.1.1" xref="S3.p1.9.m9.1.1.1.cmml">=</mo><mn id="S3.p1.9.m9.1.1.3" xref="S3.p1.9.m9.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.9.m9.1b"><apply id="S3.p1.9.m9.1.1.cmml" xref="S3.p1.9.m9.1.1"><eq id="S3.p1.9.m9.1.1.1.cmml" xref="S3.p1.9.m9.1.1.1"></eq><ci id="S3.p1.9.m9.1.1.2.cmml" xref="S3.p1.9.m9.1.1.2">𝐵</ci><cn id="S3.p1.9.m9.1.1.3.cmml" type="float" xref="S3.p1.9.m9.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m9.1c">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m9.1d">italic_B = 3.5</annotation></semantics></math>. It shows a clear exponential decay as <math alttext="e^{-\gamma s}" class="ltx_Math" display="inline" id="S3.p1.10.m10.1"><semantics id="S3.p1.10.m10.1a"><msup id="S3.p1.10.m10.1.1" xref="S3.p1.10.m10.1.1.cmml"><mi id="S3.p1.10.m10.1.1.2" xref="S3.p1.10.m10.1.1.2.cmml">e</mi><mrow id="S3.p1.10.m10.1.1.3" xref="S3.p1.10.m10.1.1.3.cmml"><mo id="S3.p1.10.m10.1.1.3a" xref="S3.p1.10.m10.1.1.3.cmml">−</mo><mrow id="S3.p1.10.m10.1.1.3.2" xref="S3.p1.10.m10.1.1.3.2.cmml"><mi id="S3.p1.10.m10.1.1.3.2.2" xref="S3.p1.10.m10.1.1.3.2.2.cmml">γ</mi><mo id="S3.p1.10.m10.1.1.3.2.1" xref="S3.p1.10.m10.1.1.3.2.1.cmml">⁢</mo><mi id="S3.p1.10.m10.1.1.3.2.3" xref="S3.p1.10.m10.1.1.3.2.3.cmml">s</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p1.10.m10.1b"><apply id="S3.p1.10.m10.1.1.cmml" xref="S3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p1.10.m10.1.1.1.cmml" xref="S3.p1.10.m10.1.1">superscript</csymbol><ci id="S3.p1.10.m10.1.1.2.cmml" xref="S3.p1.10.m10.1.1.2">𝑒</ci><apply id="S3.p1.10.m10.1.1.3.cmml" xref="S3.p1.10.m10.1.1.3"><minus id="S3.p1.10.m10.1.1.3.1.cmml" xref="S3.p1.10.m10.1.1.3"></minus><apply id="S3.p1.10.m10.1.1.3.2.cmml" xref="S3.p1.10.m10.1.1.3.2"><times id="S3.p1.10.m10.1.1.3.2.1.cmml" xref="S3.p1.10.m10.1.1.3.2.1"></times><ci id="S3.p1.10.m10.1.1.3.2.2.cmml" xref="S3.p1.10.m10.1.1.3.2.2">𝛾</ci><ci id="S3.p1.10.m10.1.1.3.2.3.cmml" xref="S3.p1.10.m10.1.1.3.2.3">𝑠</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.10.m10.1c">e^{-\gamma s}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.10.m10.1d">italic_e start_POSTSUPERSCRIPT - italic_γ italic_s end_POSTSUPERSCRIPT</annotation></semantics></math>, with decay rate <math alttext="\gamma=0.3682\pm 0.005" class="ltx_Math" display="inline" id="S3.p1.11.m11.1"><semantics id="S3.p1.11.m11.1a"><mrow id="S3.p1.11.m11.1.1" xref="S3.p1.11.m11.1.1.cmml"><mi id="S3.p1.11.m11.1.1.2" xref="S3.p1.11.m11.1.1.2.cmml">γ</mi><mo id="S3.p1.11.m11.1.1.1" xref="S3.p1.11.m11.1.1.1.cmml">=</mo><mrow id="S3.p1.11.m11.1.1.3" xref="S3.p1.11.m11.1.1.3.cmml"><mn id="S3.p1.11.m11.1.1.3.2" xref="S3.p1.11.m11.1.1.3.2.cmml">0.3682</mn><mo id="S3.p1.11.m11.1.1.3.1" xref="S3.p1.11.m11.1.1.3.1.cmml">±</mo><mn id="S3.p1.11.m11.1.1.3.3" xref="S3.p1.11.m11.1.1.3.3.cmml">0.005</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.11.m11.1b"><apply id="S3.p1.11.m11.1.1.cmml" xref="S3.p1.11.m11.1.1"><eq id="S3.p1.11.m11.1.1.1.cmml" xref="S3.p1.11.m11.1.1.1"></eq><ci id="S3.p1.11.m11.1.1.2.cmml" xref="S3.p1.11.m11.1.1.2">𝛾</ci><apply id="S3.p1.11.m11.1.1.3.cmml" xref="S3.p1.11.m11.1.1.3"><csymbol cd="latexml" id="S3.p1.11.m11.1.1.3.1.cmml" xref="S3.p1.11.m11.1.1.3.1">plus-or-minus</csymbol><cn id="S3.p1.11.m11.1.1.3.2.cmml" type="float" xref="S3.p1.11.m11.1.1.3.2">0.3682</cn><cn id="S3.p1.11.m11.1.1.3.3.cmml" type="float" xref="S3.p1.11.m11.1.1.3.3">0.005</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.11.m11.1c">\gamma=0.3682\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S3.p1.11.m11.1d">italic_γ = 0.3682 ± 0.005</annotation></semantics></math> in units of inverse time.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">We now discuss the fitting method for finding <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mi id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><ci id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_γ</annotation></semantics></math> in more detail. At early times, many trajectories escape very quickly leading to transient behavior that overestimates the decay rate. At late times, there is simply not a statistically significant number of surviving trajectories. The proper fitting region lies somewhere between these two poorly behaved regions. To address this we use a fitting method described in Ref. <cite class="ltx_cite ltx_citemacro_citep">Deshmukh <em class="ltx_emph ltx_font_italic">et al.</em>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib26" title="">2023</a></cite>. The procedure is to first generate a linear fit between every possible pair of endpoints, and then generate a histogram of the slopes from those fits. Finally, we fit a smooth probability distribution to the histogram and extract the local maximum (for the final decay rate) and standard deviation (for the error on the decay rate). This method provides two advantages: 1. It automatically computes the slope in the region of interest, without throwing away any of the dataset, while simultaneously providing a robust error measurement. 2. It automatically detects regions with multi-exponential decay, which have multiple local maxima corresponding to different decay rates.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">A problem with the Monte Carlo computation is that it does not elucidate any information about the underlying dynamics of the system. Additionally, the computation must be entirely recomputed for even a small change in parameter values. We will introduce an alternative approach using periodic orbits in the next section and use the Monte Carlo data to verify our results.</p> </div> <figure class="ltx_figure" id="S3.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="250" id="S3.F1.g1" src="extracted/6294607/monteCombinedFigureBig.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F1.10.5.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S3.F1.8.4" style="font-size:90%;">Exponential decay of the surviving (unionized) trajectories from classical trajectory Monte Carlo data at <math alttext="E=1" class="ltx_Math" display="inline" id="S3.F1.5.1.m1.1"><semantics id="S3.F1.5.1.m1.1b"><mrow id="S3.F1.5.1.m1.1.1" xref="S3.F1.5.1.m1.1.1.cmml"><mi id="S3.F1.5.1.m1.1.1.2" xref="S3.F1.5.1.m1.1.1.2.cmml">E</mi><mo id="S3.F1.5.1.m1.1.1.1" xref="S3.F1.5.1.m1.1.1.1.cmml">=</mo><mn id="S3.F1.5.1.m1.1.1.3" xref="S3.F1.5.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.5.1.m1.1c"><apply id="S3.F1.5.1.m1.1.1.cmml" xref="S3.F1.5.1.m1.1.1"><eq id="S3.F1.5.1.m1.1.1.1.cmml" xref="S3.F1.5.1.m1.1.1.1"></eq><ci id="S3.F1.5.1.m1.1.1.2.cmml" xref="S3.F1.5.1.m1.1.1.2">𝐸</ci><cn id="S3.F1.5.1.m1.1.1.3.cmml" type="integer" xref="S3.F1.5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.5.1.m1.1d">E=1</annotation><annotation encoding="application/x-llamapun" id="S3.F1.5.1.m1.1e">italic_E = 1</annotation></semantics></math> and <math alttext="B=3.5" class="ltx_Math" display="inline" id="S3.F1.6.2.m2.1"><semantics id="S3.F1.6.2.m2.1b"><mrow id="S3.F1.6.2.m2.1.1" xref="S3.F1.6.2.m2.1.1.cmml"><mi id="S3.F1.6.2.m2.1.1.2" xref="S3.F1.6.2.m2.1.1.2.cmml">B</mi><mo id="S3.F1.6.2.m2.1.1.1" xref="S3.F1.6.2.m2.1.1.1.cmml">=</mo><mn id="S3.F1.6.2.m2.1.1.3" xref="S3.F1.6.2.m2.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.6.2.m2.1c"><apply id="S3.F1.6.2.m2.1.1.cmml" xref="S3.F1.6.2.m2.1.1"><eq id="S3.F1.6.2.m2.1.1.1.cmml" xref="S3.F1.6.2.m2.1.1.1"></eq><ci id="S3.F1.6.2.m2.1.1.2.cmml" xref="S3.F1.6.2.m2.1.1.2">𝐵</ci><cn id="S3.F1.6.2.m2.1.1.3.cmml" type="float" xref="S3.F1.6.2.m2.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.6.2.m2.1d">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S3.F1.6.2.m2.1e">italic_B = 3.5</annotation></semantics></math>. Simulated data shown in black and linear fit shown in red. (a) Continuous-time simulation; <math alttext="\gamma=0.3682\pm 0.005" class="ltx_Math" display="inline" id="S3.F1.7.3.m3.1"><semantics id="S3.F1.7.3.m3.1b"><mrow id="S3.F1.7.3.m3.1.1" xref="S3.F1.7.3.m3.1.1.cmml"><mi id="S3.F1.7.3.m3.1.1.2" xref="S3.F1.7.3.m3.1.1.2.cmml">γ</mi><mo id="S3.F1.7.3.m3.1.1.1" xref="S3.F1.7.3.m3.1.1.1.cmml">=</mo><mrow id="S3.F1.7.3.m3.1.1.3" xref="S3.F1.7.3.m3.1.1.3.cmml"><mn id="S3.F1.7.3.m3.1.1.3.2" xref="S3.F1.7.3.m3.1.1.3.2.cmml">0.3682</mn><mo id="S3.F1.7.3.m3.1.1.3.1" xref="S3.F1.7.3.m3.1.1.3.1.cmml">±</mo><mn id="S3.F1.7.3.m3.1.1.3.3" xref="S3.F1.7.3.m3.1.1.3.3.cmml">0.005</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.7.3.m3.1c"><apply id="S3.F1.7.3.m3.1.1.cmml" xref="S3.F1.7.3.m3.1.1"><eq id="S3.F1.7.3.m3.1.1.1.cmml" xref="S3.F1.7.3.m3.1.1.1"></eq><ci id="S3.F1.7.3.m3.1.1.2.cmml" xref="S3.F1.7.3.m3.1.1.2">𝛾</ci><apply id="S3.F1.7.3.m3.1.1.3.cmml" xref="S3.F1.7.3.m3.1.1.3"><csymbol cd="latexml" id="S3.F1.7.3.m3.1.1.3.1.cmml" xref="S3.F1.7.3.m3.1.1.3.1">plus-or-minus</csymbol><cn id="S3.F1.7.3.m3.1.1.3.2.cmml" type="float" xref="S3.F1.7.3.m3.1.1.3.2">0.3682</cn><cn id="S3.F1.7.3.m3.1.1.3.3.cmml" type="float" xref="S3.F1.7.3.m3.1.1.3.3">0.005</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.7.3.m3.1d">\gamma=0.3682\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S3.F1.7.3.m3.1e">italic_γ = 0.3682 ± 0.005</annotation></semantics></math>. (b) Discrete-time simulation using the map; <math alttext="\gamma_{d}=0.8456\pm 0.012" class="ltx_Math" display="inline" id="S3.F1.8.4.m4.1"><semantics id="S3.F1.8.4.m4.1b"><mrow id="S3.F1.8.4.m4.1.1" xref="S3.F1.8.4.m4.1.1.cmml"><msub id="S3.F1.8.4.m4.1.1.2" xref="S3.F1.8.4.m4.1.1.2.cmml"><mi id="S3.F1.8.4.m4.1.1.2.2" xref="S3.F1.8.4.m4.1.1.2.2.cmml">γ</mi><mi id="S3.F1.8.4.m4.1.1.2.3" xref="S3.F1.8.4.m4.1.1.2.3.cmml">d</mi></msub><mo id="S3.F1.8.4.m4.1.1.1" xref="S3.F1.8.4.m4.1.1.1.cmml">=</mo><mrow id="S3.F1.8.4.m4.1.1.3" xref="S3.F1.8.4.m4.1.1.3.cmml"><mn id="S3.F1.8.4.m4.1.1.3.2" xref="S3.F1.8.4.m4.1.1.3.2.cmml">0.8456</mn><mo id="S3.F1.8.4.m4.1.1.3.1" xref="S3.F1.8.4.m4.1.1.3.1.cmml">±</mo><mn id="S3.F1.8.4.m4.1.1.3.3" xref="S3.F1.8.4.m4.1.1.3.3.cmml">0.012</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.8.4.m4.1c"><apply id="S3.F1.8.4.m4.1.1.cmml" xref="S3.F1.8.4.m4.1.1"><eq id="S3.F1.8.4.m4.1.1.1.cmml" xref="S3.F1.8.4.m4.1.1.1"></eq><apply id="S3.F1.8.4.m4.1.1.2.cmml" xref="S3.F1.8.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.F1.8.4.m4.1.1.2.1.cmml" xref="S3.F1.8.4.m4.1.1.2">subscript</csymbol><ci id="S3.F1.8.4.m4.1.1.2.2.cmml" xref="S3.F1.8.4.m4.1.1.2.2">𝛾</ci><ci id="S3.F1.8.4.m4.1.1.2.3.cmml" xref="S3.F1.8.4.m4.1.1.2.3">𝑑</ci></apply><apply id="S3.F1.8.4.m4.1.1.3.cmml" xref="S3.F1.8.4.m4.1.1.3"><csymbol cd="latexml" id="S3.F1.8.4.m4.1.1.3.1.cmml" xref="S3.F1.8.4.m4.1.1.3.1">plus-or-minus</csymbol><cn id="S3.F1.8.4.m4.1.1.3.2.cmml" type="float" xref="S3.F1.8.4.m4.1.1.3.2">0.8456</cn><cn id="S3.F1.8.4.m4.1.1.3.3.cmml" type="float" xref="S3.F1.8.4.m4.1.1.3.3">0.012</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.8.4.m4.1d">\gamma_{d}=0.8456\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S3.F1.8.4.m4.1e">italic_γ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0.8456 ± 0.012</annotation></semantics></math>.</span></figcaption> </figure> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span>Periodic Orbit Theory and Spectral Determinants</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.3">Here, we discuss an alternative technique to compute decay rates based on far fewer trajectories and a deeper understanding of the underlying electron dynamics. For details, see Ref. <cite class="ltx_cite ltx_citemacro_citep"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib3" title="">Cvitanovic <em class="ltx_emph ltx_font_italic">et al.</em>, </a></cite>. We begin by considering a general two-degree-of-freedom classical Hamiltonian system with four-dimensional phase space variable <math alttext="x" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mi id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><ci id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">italic_x</annotation></semantics></math>. We define the operator <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><ci id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">caligraphic_A</annotation></semantics></math>, which generates time-evolution by evolving a smooth state space density <math alttext="\rho\left(x,t\right)" class="ltx_Math" display="inline" id="S4.p1.3.m3.2"><semantics id="S4.p1.3.m3.2a"><mrow id="S4.p1.3.m3.2.3" xref="S4.p1.3.m3.2.3.cmml"><mi id="S4.p1.3.m3.2.3.2" xref="S4.p1.3.m3.2.3.2.cmml">ρ</mi><mo id="S4.p1.3.m3.2.3.1" xref="S4.p1.3.m3.2.3.1.cmml">⁢</mo><mrow id="S4.p1.3.m3.2.3.3.2" xref="S4.p1.3.m3.2.3.3.1.cmml"><mo id="S4.p1.3.m3.2.3.3.2.1" xref="S4.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml">x</mi><mo id="S4.p1.3.m3.2.3.3.2.2" xref="S4.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.p1.3.m3.2.2" xref="S4.p1.3.m3.2.2.cmml">t</mi><mo id="S4.p1.3.m3.2.3.3.2.3" xref="S4.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.2b"><apply id="S4.p1.3.m3.2.3.cmml" xref="S4.p1.3.m3.2.3"><times id="S4.p1.3.m3.2.3.1.cmml" xref="S4.p1.3.m3.2.3.1"></times><ci id="S4.p1.3.m3.2.3.2.cmml" xref="S4.p1.3.m3.2.3.2">𝜌</ci><interval closure="open" id="S4.p1.3.m3.2.3.3.1.cmml" xref="S4.p1.3.m3.2.3.3.2"><ci id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1">𝑥</ci><ci id="S4.p1.3.m3.2.2.cmml" xref="S4.p1.3.m3.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.2c">\rho\left(x,t\right)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.2d">italic_ρ ( italic_x , italic_t )</annotation></semantics></math> forward in time, according to the evolution equation</p> <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <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(\frac{\partial}{\partial t}-\mathcal{A}\right)\rho\left(x,t\right)=0." class="ltx_Math" display="block" id="S4.E6.m1.3"><semantics id="S4.E6.m1.3a"><mrow id="S4.E6.m1.3.3.1" xref="S4.E6.m1.3.3.1.1.cmml"><mrow id="S4.E6.m1.3.3.1.1" xref="S4.E6.m1.3.3.1.1.cmml"><mrow id="S4.E6.m1.3.3.1.1.1" xref="S4.E6.m1.3.3.1.1.1.cmml"><mrow id="S4.E6.m1.3.3.1.1.1.1.1" xref="S4.E6.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S4.E6.m1.3.3.1.1.1.1.1.2" xref="S4.E6.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E6.m1.3.3.1.1.1.1.1.1" xref="S4.E6.m1.3.3.1.1.1.1.1.1.cmml"><mfrac id="S4.E6.m1.3.3.1.1.1.1.1.1.2" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.cmml"><mo id="S4.E6.m1.3.3.1.1.1.1.1.1.2.2" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.2.cmml">∂</mo><mrow id="S4.E6.m1.3.3.1.1.1.1.1.1.2.3" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.cmml"><mo id="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.1" rspace="0em" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.1.cmml">∂</mo><mi id="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.2" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.2.cmml">t</mi></mrow></mfrac><mo id="S4.E6.m1.3.3.1.1.1.1.1.1.1" xref="S4.E6.m1.3.3.1.1.1.1.1.1.1.cmml">−</mo><mi class="ltx_font_mathcaligraphic" id="S4.E6.m1.3.3.1.1.1.1.1.1.3" xref="S4.E6.m1.3.3.1.1.1.1.1.1.3.cmml">𝒜</mi></mrow><mo id="S4.E6.m1.3.3.1.1.1.1.1.3" xref="S4.E6.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E6.m1.3.3.1.1.1.2" xref="S4.E6.m1.3.3.1.1.1.2.cmml">⁢</mo><mi id="S4.E6.m1.3.3.1.1.1.3" xref="S4.E6.m1.3.3.1.1.1.3.cmml">ρ</mi><mo id="S4.E6.m1.3.3.1.1.1.2a" xref="S4.E6.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.E6.m1.3.3.1.1.1.4.2" xref="S4.E6.m1.3.3.1.1.1.4.1.cmml"><mo id="S4.E6.m1.3.3.1.1.1.4.2.1" xref="S4.E6.m1.3.3.1.1.1.4.1.cmml">(</mo><mi id="S4.E6.m1.1.1" xref="S4.E6.m1.1.1.cmml">x</mi><mo id="S4.E6.m1.3.3.1.1.1.4.2.2" xref="S4.E6.m1.3.3.1.1.1.4.1.cmml">,</mo><mi id="S4.E6.m1.2.2" xref="S4.E6.m1.2.2.cmml">t</mi><mo id="S4.E6.m1.3.3.1.1.1.4.2.3" xref="S4.E6.m1.3.3.1.1.1.4.1.cmml">)</mo></mrow></mrow><mo id="S4.E6.m1.3.3.1.1.2" xref="S4.E6.m1.3.3.1.1.2.cmml">=</mo><mn id="S4.E6.m1.3.3.1.1.3" xref="S4.E6.m1.3.3.1.1.3.cmml">0</mn></mrow><mo id="S4.E6.m1.3.3.1.2" lspace="0em" xref="S4.E6.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.3b"><apply id="S4.E6.m1.3.3.1.1.cmml" xref="S4.E6.m1.3.3.1"><eq id="S4.E6.m1.3.3.1.1.2.cmml" xref="S4.E6.m1.3.3.1.1.2"></eq><apply id="S4.E6.m1.3.3.1.1.1.cmml" xref="S4.E6.m1.3.3.1.1.1"><times id="S4.E6.m1.3.3.1.1.1.2.cmml" xref="S4.E6.m1.3.3.1.1.1.2"></times><apply id="S4.E6.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1"><minus id="S4.E6.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.1"></minus><apply id="S4.E6.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2"><divide id="S4.E6.m1.3.3.1.1.1.1.1.1.2.1.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2"></divide><partialdiff id="S4.E6.m1.3.3.1.1.1.1.1.1.2.2.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.2"></partialdiff><apply id="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.3"><partialdiff id="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.1.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.1"></partialdiff><ci id="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.2.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.2.3.2">𝑡</ci></apply></apply><ci id="S4.E6.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S4.E6.m1.3.3.1.1.1.1.1.1.3">𝒜</ci></apply><ci id="S4.E6.m1.3.3.1.1.1.3.cmml" xref="S4.E6.m1.3.3.1.1.1.3">𝜌</ci><interval closure="open" id="S4.E6.m1.3.3.1.1.1.4.1.cmml" xref="S4.E6.m1.3.3.1.1.1.4.2"><ci id="S4.E6.m1.1.1.cmml" xref="S4.E6.m1.1.1">𝑥</ci><ci id="S4.E6.m1.2.2.cmml" xref="S4.E6.m1.2.2">𝑡</ci></interval></apply><cn id="S4.E6.m1.3.3.1.1.3.cmml" type="integer" xref="S4.E6.m1.3.3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.3c">\left(\frac{\partial}{\partial t}-\mathcal{A}\right)\rho\left(x,t\right)=0.</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.3d">( divide start_ARG ∂ end_ARG start_ARG ∂ italic_t end_ARG - caligraphic_A ) italic_ρ ( italic_x , italic_t ) = 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="S4.p1.6">This motivates the definition of the spectral determinant <math alttext="\text{det}\left(s-\mathcal{A}\right)" class="ltx_Math" display="inline" id="S4.p1.4.m1.1"><semantics id="S4.p1.4.m1.1a"><mrow id="S4.p1.4.m1.1.1" xref="S4.p1.4.m1.1.1.cmml"><mtext id="S4.p1.4.m1.1.1.3" xref="S4.p1.4.m1.1.1.3a.cmml">det</mtext><mo id="S4.p1.4.m1.1.1.2" xref="S4.p1.4.m1.1.1.2.cmml">⁢</mo><mrow id="S4.p1.4.m1.1.1.1.1" xref="S4.p1.4.m1.1.1.1.1.1.cmml"><mo id="S4.p1.4.m1.1.1.1.1.2" xref="S4.p1.4.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.p1.4.m1.1.1.1.1.1" xref="S4.p1.4.m1.1.1.1.1.1.cmml"><mi id="S4.p1.4.m1.1.1.1.1.1.2" xref="S4.p1.4.m1.1.1.1.1.1.2.cmml">s</mi><mo id="S4.p1.4.m1.1.1.1.1.1.1" xref="S4.p1.4.m1.1.1.1.1.1.1.cmml">−</mo><mi class="ltx_font_mathcaligraphic" id="S4.p1.4.m1.1.1.1.1.1.3" xref="S4.p1.4.m1.1.1.1.1.1.3.cmml">𝒜</mi></mrow><mo id="S4.p1.4.m1.1.1.1.1.3" xref="S4.p1.4.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.4.m1.1b"><apply id="S4.p1.4.m1.1.1.cmml" xref="S4.p1.4.m1.1.1"><times id="S4.p1.4.m1.1.1.2.cmml" xref="S4.p1.4.m1.1.1.2"></times><ci id="S4.p1.4.m1.1.1.3a.cmml" xref="S4.p1.4.m1.1.1.3"><mtext id="S4.p1.4.m1.1.1.3.cmml" xref="S4.p1.4.m1.1.1.3">det</mtext></ci><apply id="S4.p1.4.m1.1.1.1.1.1.cmml" xref="S4.p1.4.m1.1.1.1.1"><minus id="S4.p1.4.m1.1.1.1.1.1.1.cmml" xref="S4.p1.4.m1.1.1.1.1.1.1"></minus><ci id="S4.p1.4.m1.1.1.1.1.1.2.cmml" xref="S4.p1.4.m1.1.1.1.1.1.2">𝑠</ci><ci id="S4.p1.4.m1.1.1.1.1.1.3.cmml" xref="S4.p1.4.m1.1.1.1.1.1.3">𝒜</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m1.1c">\text{det}\left(s-\mathcal{A}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m1.1d">det ( italic_s - caligraphic_A )</annotation></semantics></math>, which we view as a function of <math alttext="s" class="ltx_Math" display="inline" id="S4.p1.5.m2.1"><semantics id="S4.p1.5.m2.1a"><mi id="S4.p1.5.m2.1.1" xref="S4.p1.5.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S4.p1.5.m2.1b"><ci id="S4.p1.5.m2.1.1.cmml" xref="S4.p1.5.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m2.1d">italic_s</annotation></semantics></math>. The zeros of this determinant give the spectrum of the time-evolution generator <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S4.p1.6.m3.1"><semantics id="S4.p1.6.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S4.p1.6.m3.1.1" xref="S4.p1.6.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S4.p1.6.m3.1b"><ci id="S4.p1.6.m3.1.1.cmml" xref="S4.p1.6.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.m3.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m3.1d">caligraphic_A</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib3" title="">Cvitanovic <em class="ltx_emph ltx_font_italic">et al.</em> </a></cite>, and the smallest eigenvalue determines the long-time escape rate of the system. Furthermore, this spectral determinant can be computed as a sum over the periodic orbits of the dynamical system as follows</p> <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <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="\text{det}(s-\mathcal{A})=\text{exp}\left(-\sum_{p}\sum^{\infty}_{r=1}\frac{1}% {r}\frac{e^{-srT_{p}}}{|\Lambda_{p}|^{\frac{1}{2}}}\right)" class="ltx_Math" display="block" id="S4.E7.m1.3"><semantics id="S4.E7.m1.3a"><mrow id="S4.E7.m1.3.3" xref="S4.E7.m1.3.3.cmml"><mrow id="S4.E7.m1.2.2.1" xref="S4.E7.m1.2.2.1.cmml"><mtext id="S4.E7.m1.2.2.1.3" xref="S4.E7.m1.2.2.1.3a.cmml">det</mtext><mo id="S4.E7.m1.2.2.1.2" xref="S4.E7.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.E7.m1.2.2.1.1.1" xref="S4.E7.m1.2.2.1.1.1.1.cmml"><mo id="S4.E7.m1.2.2.1.1.1.2" stretchy="false" xref="S4.E7.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.E7.m1.2.2.1.1.1.1" xref="S4.E7.m1.2.2.1.1.1.1.cmml"><mi id="S4.E7.m1.2.2.1.1.1.1.2" xref="S4.E7.m1.2.2.1.1.1.1.2.cmml">s</mi><mo id="S4.E7.m1.2.2.1.1.1.1.1" xref="S4.E7.m1.2.2.1.1.1.1.1.cmml">−</mo><mi class="ltx_font_mathcaligraphic" id="S4.E7.m1.2.2.1.1.1.1.3" xref="S4.E7.m1.2.2.1.1.1.1.3.cmml">𝒜</mi></mrow><mo id="S4.E7.m1.2.2.1.1.1.3" stretchy="false" xref="S4.E7.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.3.3.3" xref="S4.E7.m1.3.3.3.cmml">=</mo><mrow id="S4.E7.m1.3.3.2" xref="S4.E7.m1.3.3.2.cmml"><mtext id="S4.E7.m1.3.3.2.3" xref="S4.E7.m1.3.3.2.3a.cmml">exp</mtext><mo id="S4.E7.m1.3.3.2.2" xref="S4.E7.m1.3.3.2.2.cmml">⁢</mo><mrow id="S4.E7.m1.3.3.2.1.1" xref="S4.E7.m1.3.3.2.1.1.1.cmml"><mo id="S4.E7.m1.3.3.2.1.1.2" xref="S4.E7.m1.3.3.2.1.1.1.cmml">(</mo><mrow id="S4.E7.m1.3.3.2.1.1.1" xref="S4.E7.m1.3.3.2.1.1.1.cmml"><mo id="S4.E7.m1.3.3.2.1.1.1a" xref="S4.E7.m1.3.3.2.1.1.1.cmml">−</mo><mrow id="S4.E7.m1.3.3.2.1.1.1.2" xref="S4.E7.m1.3.3.2.1.1.1.2.cmml"><munder 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id="S4.E7.m1.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.1.1.1.1.1.1.2">Λ</ci><ci id="S4.E7.m1.1.1.1.1.1.1.3.cmml" xref="S4.E7.m1.1.1.1.1.1.1.3">𝑝</ci></apply></apply><apply id="S4.E7.m1.1.1.1.3.cmml" xref="S4.E7.m1.1.1.1.3"><divide id="S4.E7.m1.1.1.1.3.1.cmml" xref="S4.E7.m1.1.1.1.3"></divide><cn id="S4.E7.m1.1.1.1.3.2.cmml" type="integer" xref="S4.E7.m1.1.1.1.3.2">1</cn><cn id="S4.E7.m1.1.1.1.3.3.cmml" type="integer" xref="S4.E7.m1.1.1.1.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.3c">\text{det}(s-\mathcal{A})=\text{exp}\left(-\sum_{p}\sum^{\infty}_{r=1}\frac{1}% {r}\frac{e^{-srT_{p}}}{|\Lambda_{p}|^{\frac{1}{2}}}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.3d">det ( italic_s - caligraphic_A ) = exp ( - ∑ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r = 1 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_r end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_s italic_r italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG | roman_Λ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT 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">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p1.14">where,</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="\left|\Lambda_{p}\right|=\left|\text{det}\left(1-M_{p}^{r}\right)\right|=\left% 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encoding="application/x-tex" id="S4.E8.m1.1c">\left|\Lambda_{p}\right|=\left|\text{det}\left(1-M_{p}^{r}\right)\right|=\left% |\left(1-\lambda_{p}^{r}\right)\left(1-\lambda_{p}^{-r}\right)\right|.</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.1d">| roman_Λ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT | = | det ( 1 - italic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) | = | ( 1 - italic_λ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) ( 1 - italic_λ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - italic_r 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> <p class="ltx_p" id="S4.p1.13">Here <math alttext="p" class="ltx_Math" display="inline" id="S4.p1.7.m1.1"><semantics id="S4.p1.7.m1.1a"><mi id="S4.p1.7.m1.1.1" xref="S4.p1.7.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.p1.7.m1.1b"><ci id="S4.p1.7.m1.1.1.cmml" xref="S4.p1.7.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m1.1d">italic_p</annotation></semantics></math> is an index that runs through all distinct prime periodic orbits, and <math alttext="r" class="ltx_Math" display="inline" id="S4.p1.8.m2.1"><semantics id="S4.p1.8.m2.1a"><mi id="S4.p1.8.m2.1.1" xref="S4.p1.8.m2.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.p1.8.m2.1b"><ci id="S4.p1.8.m2.1.1.cmml" xref="S4.p1.8.m2.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.8.m2.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.p1.8.m2.1d">italic_r</annotation></semantics></math> is an index specifying the number of times each prime orbit is retraced. (A prime orbit is one that does not retrace itself.) The matrix <math alttext="M_{p}" class="ltx_Math" display="inline" id="S4.p1.9.m3.1"><semantics id="S4.p1.9.m3.1a"><msub id="S4.p1.9.m3.1.1" xref="S4.p1.9.m3.1.1.cmml"><mi id="S4.p1.9.m3.1.1.2" xref="S4.p1.9.m3.1.1.2.cmml">M</mi><mi id="S4.p1.9.m3.1.1.3" xref="S4.p1.9.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.9.m3.1b"><apply id="S4.p1.9.m3.1.1.cmml" xref="S4.p1.9.m3.1.1"><csymbol cd="ambiguous" id="S4.p1.9.m3.1.1.1.cmml" xref="S4.p1.9.m3.1.1">subscript</csymbol><ci id="S4.p1.9.m3.1.1.2.cmml" xref="S4.p1.9.m3.1.1.2">𝑀</ci><ci id="S4.p1.9.m3.1.1.3.cmml" xref="S4.p1.9.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.9.m3.1c">M_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.9.m3.1d">italic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is the <math alttext="2\times 2" class="ltx_Math" display="inline" id="S4.p1.10.m4.1"><semantics id="S4.p1.10.m4.1a"><mrow id="S4.p1.10.m4.1.1" xref="S4.p1.10.m4.1.1.cmml"><mn id="S4.p1.10.m4.1.1.2" xref="S4.p1.10.m4.1.1.2.cmml">2</mn><mo id="S4.p1.10.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.p1.10.m4.1.1.1.cmml">×</mo><mn id="S4.p1.10.m4.1.1.3" xref="S4.p1.10.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.10.m4.1b"><apply id="S4.p1.10.m4.1.1.cmml" xref="S4.p1.10.m4.1.1"><times id="S4.p1.10.m4.1.1.1.cmml" xref="S4.p1.10.m4.1.1.1"></times><cn id="S4.p1.10.m4.1.1.2.cmml" type="integer" xref="S4.p1.10.m4.1.1.2">2</cn><cn id="S4.p1.10.m4.1.1.3.cmml" type="integer" xref="S4.p1.10.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.10.m4.1c">2\times 2</annotation><annotation encoding="application/x-llamapun" id="S4.p1.10.m4.1d">2 × 2</annotation></semantics></math> linearization of the prime orbit over one period (i.e. the monodromy matrix), and <math alttext="T_{p}" class="ltx_Math" display="inline" id="S4.p1.11.m5.1"><semantics id="S4.p1.11.m5.1a"><msub id="S4.p1.11.m5.1.1" xref="S4.p1.11.m5.1.1.cmml"><mi id="S4.p1.11.m5.1.1.2" xref="S4.p1.11.m5.1.1.2.cmml">T</mi><mi id="S4.p1.11.m5.1.1.3" xref="S4.p1.11.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.11.m5.1b"><apply id="S4.p1.11.m5.1.1.cmml" xref="S4.p1.11.m5.1.1"><csymbol cd="ambiguous" id="S4.p1.11.m5.1.1.1.cmml" xref="S4.p1.11.m5.1.1">subscript</csymbol><ci id="S4.p1.11.m5.1.1.2.cmml" xref="S4.p1.11.m5.1.1.2">𝑇</ci><ci id="S4.p1.11.m5.1.1.3.cmml" xref="S4.p1.11.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.11.m5.1c">T_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.11.m5.1d">italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is its continuous-time period. The largest eigenvalue <math alttext="\lambda_{p}" class="ltx_Math" display="inline" id="S4.p1.12.m6.1"><semantics id="S4.p1.12.m6.1a"><msub id="S4.p1.12.m6.1.1" xref="S4.p1.12.m6.1.1.cmml"><mi id="S4.p1.12.m6.1.1.2" xref="S4.p1.12.m6.1.1.2.cmml">λ</mi><mi id="S4.p1.12.m6.1.1.3" xref="S4.p1.12.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.12.m6.1b"><apply id="S4.p1.12.m6.1.1.cmml" xref="S4.p1.12.m6.1.1"><csymbol cd="ambiguous" id="S4.p1.12.m6.1.1.1.cmml" xref="S4.p1.12.m6.1.1">subscript</csymbol><ci id="S4.p1.12.m6.1.1.2.cmml" xref="S4.p1.12.m6.1.1.2">𝜆</ci><ci id="S4.p1.12.m6.1.1.3.cmml" xref="S4.p1.12.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.12.m6.1c">\lambda_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.12.m6.1d">italic_λ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="M_{p}" class="ltx_Math" display="inline" id="S4.p1.13.m7.1"><semantics id="S4.p1.13.m7.1a"><msub id="S4.p1.13.m7.1.1" xref="S4.p1.13.m7.1.1.cmml"><mi id="S4.p1.13.m7.1.1.2" xref="S4.p1.13.m7.1.1.2.cmml">M</mi><mi id="S4.p1.13.m7.1.1.3" xref="S4.p1.13.m7.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.13.m7.1b"><apply id="S4.p1.13.m7.1.1.cmml" xref="S4.p1.13.m7.1.1"><csymbol cd="ambiguous" id="S4.p1.13.m7.1.1.1.cmml" xref="S4.p1.13.m7.1.1">subscript</csymbol><ci id="S4.p1.13.m7.1.1.2.cmml" xref="S4.p1.13.m7.1.1.2">𝑀</ci><ci id="S4.p1.13.m7.1.1.3.cmml" xref="S4.p1.13.m7.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.13.m7.1c">M_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.13.m7.1d">italic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is used to compute the determinant in the denominator. The above periodic orbit sum can be physically interpreted as follows. The set of all non-escaping points in phase space can be densely filled by the periodic orbits. Any long-time escaping trajectory will shadow these orbits as it escapes, shifting from the neighborhood of one unstable periodic orbit to another. Thus properly averaging over all periodic orbits contains the same global information as the eigenvalue spectrum itself.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">The periodic orbit sum in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S4.E7" title="In IV Periodic Orbit Theory and Spectral Determinants ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">7</span></a>) converges absolutely in the limit of including all periodic orbits. In practice, of course, only a finite number of orbits can be computed. Thus, to get sufficient accuracy, we compute all periodic orbits up to a reasonably high period. Computing such a large set of orbits in the full four-dimensional phase space is very challenging, so in the next section we will introduce a two-dimensional surface of section that we will use to find periodic orbits of the associated discrete mapping. The discrete orbits can be integrated to give the full continuous orbits needed here. However, as an intermediate step, it is useful to consider the discrete map in its own right, and to study the escape rate of this map in terms of a periodic orbit sum, as discussed next.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.3">As in the continuous case, the zeros of a spectral determinant describe the escape rate of a discrete two-dimensional map. Now, however, the spectral determinant is written as a function of <math alttext="z" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mi id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><ci id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_z</annotation></semantics></math> in the form <math alttext="\text{det}\left(1-z\mathcal{L}\right)" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mrow id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml"><mtext id="S4.p3.2.m2.1.1.3" xref="S4.p3.2.m2.1.1.3a.cmml">det</mtext><mo id="S4.p3.2.m2.1.1.2" xref="S4.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S4.p3.2.m2.1.1.1.1" xref="S4.p3.2.m2.1.1.1.1.1.cmml"><mo id="S4.p3.2.m2.1.1.1.1.2" xref="S4.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S4.p3.2.m2.1.1.1.1.1" xref="S4.p3.2.m2.1.1.1.1.1.cmml"><mn id="S4.p3.2.m2.1.1.1.1.1.2" xref="S4.p3.2.m2.1.1.1.1.1.2.cmml">1</mn><mo id="S4.p3.2.m2.1.1.1.1.1.1" xref="S4.p3.2.m2.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.p3.2.m2.1.1.1.1.1.3" xref="S4.p3.2.m2.1.1.1.1.1.3.cmml"><mi id="S4.p3.2.m2.1.1.1.1.1.3.2" xref="S4.p3.2.m2.1.1.1.1.1.3.2.cmml">z</mi><mo id="S4.p3.2.m2.1.1.1.1.1.3.1" xref="S4.p3.2.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.p3.2.m2.1.1.1.1.1.3.3" xref="S4.p3.2.m2.1.1.1.1.1.3.3.cmml">ℒ</mi></mrow></mrow><mo id="S4.p3.2.m2.1.1.1.1.3" xref="S4.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><apply id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1"><times id="S4.p3.2.m2.1.1.2.cmml" xref="S4.p3.2.m2.1.1.2"></times><ci id="S4.p3.2.m2.1.1.3a.cmml" xref="S4.p3.2.m2.1.1.3"><mtext id="S4.p3.2.m2.1.1.3.cmml" xref="S4.p3.2.m2.1.1.3">det</mtext></ci><apply id="S4.p3.2.m2.1.1.1.1.1.cmml" xref="S4.p3.2.m2.1.1.1.1"><minus id="S4.p3.2.m2.1.1.1.1.1.1.cmml" xref="S4.p3.2.m2.1.1.1.1.1.1"></minus><cn id="S4.p3.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S4.p3.2.m2.1.1.1.1.1.2">1</cn><apply id="S4.p3.2.m2.1.1.1.1.1.3.cmml" xref="S4.p3.2.m2.1.1.1.1.1.3"><times id="S4.p3.2.m2.1.1.1.1.1.3.1.cmml" xref="S4.p3.2.m2.1.1.1.1.1.3.1"></times><ci id="S4.p3.2.m2.1.1.1.1.1.3.2.cmml" xref="S4.p3.2.m2.1.1.1.1.1.3.2">𝑧</ci><ci id="S4.p3.2.m2.1.1.1.1.1.3.3.cmml" xref="S4.p3.2.m2.1.1.1.1.1.3.3">ℒ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">\text{det}\left(1-z\mathcal{L}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">det ( 1 - italic_z caligraphic_L )</annotation></semantics></math>, where <math alttext="\mathcal{L}" class="ltx_Math" display="inline" id="S4.p3.3.m3.1"><semantics id="S4.p3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S4.p3.3.m3.1.1" xref="S4.p3.3.m3.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S4.p3.3.m3.1b"><ci id="S4.p3.3.m3.1.1.cmml" xref="S4.p3.3.m3.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m3.1c">\mathcal{L}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m3.1d">caligraphic_L</annotation></semantics></math> is the discrete time density evolution operator acting on densities in phase space. This spectral determinant is related to periodic orbits in a similar way as the continuous case given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.E9"> <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="\text{det}(1-\textit{z}\mathcal{L})=\text{exp}\left(-\sum_{p}\sum^{\infty}_{r=% 1}\frac{1}{r}\frac{{z^{rn_{p}}}}{|\Lambda_{p}|^{\frac{1}{2}}}\right)," class="ltx_Math" display="block" id="S4.E9.m1.2"><semantics id="S4.E9.m1.2a"><mrow id="S4.E9.m1.2.2.1" xref="S4.E9.m1.2.2.1.1.cmml"><mrow id="S4.E9.m1.2.2.1.1" xref="S4.E9.m1.2.2.1.1.cmml"><mrow id="S4.E9.m1.2.2.1.1.1" xref="S4.E9.m1.2.2.1.1.1.cmml"><mtext id="S4.E9.m1.2.2.1.1.1.3" xref="S4.E9.m1.2.2.1.1.1.3a.cmml">det</mtext><mo id="S4.E9.m1.2.2.1.1.1.2" xref="S4.E9.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.E9.m1.2.2.1.1.1.1.1" xref="S4.E9.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S4.E9.m1.2.2.1.1.1.1.1.2" 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xref="S4.E9.m1.1.1.1.1.1.1.3">𝑝</ci></apply></apply><apply id="S4.E9.m1.1.1.1.3.cmml" xref="S4.E9.m1.1.1.1.3"><divide id="S4.E9.m1.1.1.1.3.1.cmml" xref="S4.E9.m1.1.1.1.3"></divide><cn id="S4.E9.m1.1.1.1.3.2.cmml" type="integer" xref="S4.E9.m1.1.1.1.3.2">1</cn><cn id="S4.E9.m1.1.1.1.3.3.cmml" type="integer" xref="S4.E9.m1.1.1.1.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9.m1.2c">\text{det}(1-\textit{z}\mathcal{L})=\text{exp}\left(-\sum_{p}\sum^{\infty}_{r=% 1}\frac{1}{r}\frac{{z^{rn_{p}}}}{|\Lambda_{p}|^{\frac{1}{2}}}\right),</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m1.2d">det ( 1 - z caligraphic_L ) = exp ( - ∑ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r = 1 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_r end_ARG divide start_ARG italic_z start_POSTSUPERSCRIPT italic_r italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG | roman_Λ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT 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="S4.p3.8">where <math alttext="n_{p}" class="ltx_Math" display="inline" id="S4.p3.4.m1.1"><semantics id="S4.p3.4.m1.1a"><msub id="S4.p3.4.m1.1.1" xref="S4.p3.4.m1.1.1.cmml"><mi id="S4.p3.4.m1.1.1.2" xref="S4.p3.4.m1.1.1.2.cmml">n</mi><mi id="S4.p3.4.m1.1.1.3" xref="S4.p3.4.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p3.4.m1.1b"><apply id="S4.p3.4.m1.1.1.cmml" xref="S4.p3.4.m1.1.1"><csymbol cd="ambiguous" id="S4.p3.4.m1.1.1.1.cmml" xref="S4.p3.4.m1.1.1">subscript</csymbol><ci id="S4.p3.4.m1.1.1.2.cmml" xref="S4.p3.4.m1.1.1.2">𝑛</ci><ci id="S4.p3.4.m1.1.1.3.cmml" xref="S4.p3.4.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.m1.1c">n_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.m1.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is the discrete period of an orbit. The discrete and continuous cases can be related by letting <math alttext="z=e^{-s}" class="ltx_Math" display="inline" id="S4.p3.5.m2.1"><semantics id="S4.p3.5.m2.1a"><mrow id="S4.p3.5.m2.1.1" xref="S4.p3.5.m2.1.1.cmml"><mi id="S4.p3.5.m2.1.1.2" xref="S4.p3.5.m2.1.1.2.cmml">z</mi><mo id="S4.p3.5.m2.1.1.1" xref="S4.p3.5.m2.1.1.1.cmml">=</mo><msup id="S4.p3.5.m2.1.1.3" xref="S4.p3.5.m2.1.1.3.cmml"><mi id="S4.p3.5.m2.1.1.3.2" xref="S4.p3.5.m2.1.1.3.2.cmml">e</mi><mrow id="S4.p3.5.m2.1.1.3.3" xref="S4.p3.5.m2.1.1.3.3.cmml"><mo id="S4.p3.5.m2.1.1.3.3a" xref="S4.p3.5.m2.1.1.3.3.cmml">−</mo><mi id="S4.p3.5.m2.1.1.3.3.2" xref="S4.p3.5.m2.1.1.3.3.2.cmml">s</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.5.m2.1b"><apply id="S4.p3.5.m2.1.1.cmml" xref="S4.p3.5.m2.1.1"><eq id="S4.p3.5.m2.1.1.1.cmml" xref="S4.p3.5.m2.1.1.1"></eq><ci id="S4.p3.5.m2.1.1.2.cmml" xref="S4.p3.5.m2.1.1.2">𝑧</ci><apply id="S4.p3.5.m2.1.1.3.cmml" xref="S4.p3.5.m2.1.1.3"><csymbol cd="ambiguous" id="S4.p3.5.m2.1.1.3.1.cmml" xref="S4.p3.5.m2.1.1.3">superscript</csymbol><ci id="S4.p3.5.m2.1.1.3.2.cmml" xref="S4.p3.5.m2.1.1.3.2">𝑒</ci><apply id="S4.p3.5.m2.1.1.3.3.cmml" xref="S4.p3.5.m2.1.1.3.3"><minus id="S4.p3.5.m2.1.1.3.3.1.cmml" xref="S4.p3.5.m2.1.1.3.3"></minus><ci id="S4.p3.5.m2.1.1.3.3.2.cmml" xref="S4.p3.5.m2.1.1.3.3.2">𝑠</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.5.m2.1c">z=e^{-s}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.5.m2.1d">italic_z = italic_e start_POSTSUPERSCRIPT - italic_s end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="n_{p}=T_{p}" class="ltx_Math" display="inline" id="S4.p3.6.m3.1"><semantics id="S4.p3.6.m3.1a"><mrow id="S4.p3.6.m3.1.1" xref="S4.p3.6.m3.1.1.cmml"><msub id="S4.p3.6.m3.1.1.2" xref="S4.p3.6.m3.1.1.2.cmml"><mi id="S4.p3.6.m3.1.1.2.2" xref="S4.p3.6.m3.1.1.2.2.cmml">n</mi><mi id="S4.p3.6.m3.1.1.2.3" xref="S4.p3.6.m3.1.1.2.3.cmml">p</mi></msub><mo id="S4.p3.6.m3.1.1.1" xref="S4.p3.6.m3.1.1.1.cmml">=</mo><msub id="S4.p3.6.m3.1.1.3" xref="S4.p3.6.m3.1.1.3.cmml"><mi id="S4.p3.6.m3.1.1.3.2" xref="S4.p3.6.m3.1.1.3.2.cmml">T</mi><mi id="S4.p3.6.m3.1.1.3.3" xref="S4.p3.6.m3.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.6.m3.1b"><apply id="S4.p3.6.m3.1.1.cmml" xref="S4.p3.6.m3.1.1"><eq id="S4.p3.6.m3.1.1.1.cmml" xref="S4.p3.6.m3.1.1.1"></eq><apply id="S4.p3.6.m3.1.1.2.cmml" xref="S4.p3.6.m3.1.1.2"><csymbol cd="ambiguous" id="S4.p3.6.m3.1.1.2.1.cmml" xref="S4.p3.6.m3.1.1.2">subscript</csymbol><ci id="S4.p3.6.m3.1.1.2.2.cmml" xref="S4.p3.6.m3.1.1.2.2">𝑛</ci><ci id="S4.p3.6.m3.1.1.2.3.cmml" xref="S4.p3.6.m3.1.1.2.3">𝑝</ci></apply><apply id="S4.p3.6.m3.1.1.3.cmml" xref="S4.p3.6.m3.1.1.3"><csymbol cd="ambiguous" id="S4.p3.6.m3.1.1.3.1.cmml" xref="S4.p3.6.m3.1.1.3">subscript</csymbol><ci id="S4.p3.6.m3.1.1.3.2.cmml" xref="S4.p3.6.m3.1.1.3.2">𝑇</ci><ci id="S4.p3.6.m3.1.1.3.3.cmml" xref="S4.p3.6.m3.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.6.m3.1c">n_{p}=T_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.6.m3.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. This change of variables represents moving from using discrete iterates <math alttext="n_{p}" class="ltx_Math" display="inline" id="S4.p3.7.m4.1"><semantics id="S4.p3.7.m4.1a"><msub id="S4.p3.7.m4.1.1" xref="S4.p3.7.m4.1.1.cmml"><mi id="S4.p3.7.m4.1.1.2" xref="S4.p3.7.m4.1.1.2.cmml">n</mi><mi id="S4.p3.7.m4.1.1.3" xref="S4.p3.7.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p3.7.m4.1b"><apply id="S4.p3.7.m4.1.1.cmml" xref="S4.p3.7.m4.1.1"><csymbol cd="ambiguous" id="S4.p3.7.m4.1.1.1.cmml" xref="S4.p3.7.m4.1.1">subscript</csymbol><ci id="S4.p3.7.m4.1.1.2.cmml" xref="S4.p3.7.m4.1.1.2">𝑛</ci><ci id="S4.p3.7.m4.1.1.3.cmml" xref="S4.p3.7.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.7.m4.1c">n_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.7.m4.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> to continuous time <math alttext="T_{p}" class="ltx_Math" display="inline" id="S4.p3.8.m5.1"><semantics id="S4.p3.8.m5.1a"><msub id="S4.p3.8.m5.1.1" xref="S4.p3.8.m5.1.1.cmml"><mi id="S4.p3.8.m5.1.1.2" xref="S4.p3.8.m5.1.1.2.cmml">T</mi><mi id="S4.p3.8.m5.1.1.3" xref="S4.p3.8.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p3.8.m5.1b"><apply id="S4.p3.8.m5.1.1.cmml" xref="S4.p3.8.m5.1.1"><csymbol cd="ambiguous" id="S4.p3.8.m5.1.1.1.cmml" xref="S4.p3.8.m5.1.1">subscript</csymbol><ci id="S4.p3.8.m5.1.1.2.cmml" xref="S4.p3.8.m5.1.1.2">𝑇</ci><ci id="S4.p3.8.m5.1.1.3.cmml" xref="S4.p3.8.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.8.m5.1c">T_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.8.m5.1d">italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.2">Computing zeros of the spectral determinant directly from Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S4.E7" title="In IV Periodic Orbit Theory and Spectral Determinants ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">7</span></a>) or Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S4.E9" title="In IV Periodic Orbit Theory and Spectral Determinants ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">9</span></a>) is not trivial. Here we will provide a practical description of how to perform this computation. A detailed derivation of the following method is described in Ref (<cite class="ltx_cite ltx_citemacro_citep"><a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib3" title="">Cvitanovic <em class="ltx_emph ltx_font_italic">et al.</em>, </a></cite>), and a concise derivation of the discrete case is given in Ref (<cite class="ltx_cite ltx_citemacro_citep">Sattari and Mitchell, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib15" title="">2017</a></cite>). First, the spectral determinant up to discrete period <math alttext="N" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><mi id="S4.p4.1.m1.1.1" xref="S4.p4.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.p4.1.m1.1b"><ci id="S4.p4.1.m1.1.1.cmml" xref="S4.p4.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">italic_N</annotation></semantics></math> is written as a power series expansion with coefficients <math alttext="Q_{n}" class="ltx_Math" display="inline" id="S4.p4.2.m2.1"><semantics id="S4.p4.2.m2.1a"><msub id="S4.p4.2.m2.1.1" xref="S4.p4.2.m2.1.1.cmml"><mi id="S4.p4.2.m2.1.1.2" xref="S4.p4.2.m2.1.1.2.cmml">Q</mi><mi id="S4.p4.2.m2.1.1.3" xref="S4.p4.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p4.2.m2.1b"><apply id="S4.p4.2.m2.1.1.cmml" xref="S4.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p4.2.m2.1.1.1.cmml" xref="S4.p4.2.m2.1.1">subscript</csymbol><ci id="S4.p4.2.m2.1.1.2.cmml" xref="S4.p4.2.m2.1.1.2">𝑄</ci><ci id="S4.p4.2.m2.1.1.3.cmml" xref="S4.p4.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.2.m2.1c">Q_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.2.m2.1d">italic_Q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.E10"> <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.\text{det}\left(s-\mathcal{A}\right)\right|_{N}=1-\sum_{n=1}^{N}Q_{n}z^{% n}." class="ltx_Math" display="block" id="S4.E10.m1.2"><semantics id="S4.E10.m1.2a"><mrow id="S4.E10.m1.2.2.1" xref="S4.E10.m1.2.2.1.1.cmml"><mrow id="S4.E10.m1.2.2.1.1" xref="S4.E10.m1.2.2.1.1.cmml"><msub id="S4.E10.m1.2.2.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.2.cmml"><mrow id="S4.E10.m1.2.2.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.2.cmml"><mrow id="S4.E10.m1.2.2.1.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.1.1.1.cmml"><mtext id="S4.E10.m1.2.2.1.1.1.1.1.1.3" xref="S4.E10.m1.2.2.1.1.1.1.1.1.3a.cmml">det</mtext><mo id="S4.E10.m1.2.2.1.1.1.1.1.1.2" xref="S4.E10.m1.2.2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.2" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.2" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">s</mi><mo id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi class="ltx_font_mathcaligraphic" id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.3" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.3.cmml">𝒜</mi></mrow><mo 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xref="S4.E10.m1.2.2.1.1.3.3.1.2.3.1.cmml">=</mo><mn id="S4.E10.m1.2.2.1.1.3.3.1.2.3.3" xref="S4.E10.m1.2.2.1.1.3.3.1.2.3.3.cmml">1</mn></mrow><mi id="S4.E10.m1.2.2.1.1.3.3.1.3" xref="S4.E10.m1.2.2.1.1.3.3.1.3.cmml">N</mi></munderover><mrow id="S4.E10.m1.2.2.1.1.3.3.2" xref="S4.E10.m1.2.2.1.1.3.3.2.cmml"><msub id="S4.E10.m1.2.2.1.1.3.3.2.2" xref="S4.E10.m1.2.2.1.1.3.3.2.2.cmml"><mi id="S4.E10.m1.2.2.1.1.3.3.2.2.2" xref="S4.E10.m1.2.2.1.1.3.3.2.2.2.cmml">Q</mi><mi id="S4.E10.m1.2.2.1.1.3.3.2.2.3" xref="S4.E10.m1.2.2.1.1.3.3.2.2.3.cmml">n</mi></msub><mo id="S4.E10.m1.2.2.1.1.3.3.2.1" xref="S4.E10.m1.2.2.1.1.3.3.2.1.cmml">⁢</mo><msup id="S4.E10.m1.2.2.1.1.3.3.2.3" xref="S4.E10.m1.2.2.1.1.3.3.2.3.cmml"><mi id="S4.E10.m1.2.2.1.1.3.3.2.3.2" xref="S4.E10.m1.2.2.1.1.3.3.2.3.2.cmml">z</mi><mi id="S4.E10.m1.2.2.1.1.3.3.2.3.3" xref="S4.E10.m1.2.2.1.1.3.3.2.3.3.cmml">n</mi></msup></mrow></mrow></mrow></mrow><mo id="S4.E10.m1.2.2.1.2" lspace="0em" xref="S4.E10.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E10.m1.2b"><apply id="S4.E10.m1.2.2.1.1.cmml" xref="S4.E10.m1.2.2.1"><eq id="S4.E10.m1.2.2.1.1.2.cmml" xref="S4.E10.m1.2.2.1.1.2"></eq><apply id="S4.E10.m1.2.2.1.1.1.2.cmml" xref="S4.E10.m1.2.2.1.1.1.1"><csymbol cd="latexml" id="S4.E10.m1.2.2.1.1.1.2.1.cmml" xref="S4.E10.m1.2.2.1.1.1.1.1.2">evaluated-at</csymbol><apply id="S4.E10.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E10.m1.2.2.1.1.1.1.1.1"><times id="S4.E10.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E10.m1.2.2.1.1.1.1.1.1.2"></times><ci id="S4.E10.m1.2.2.1.1.1.1.1.1.3a.cmml" xref="S4.E10.m1.2.2.1.1.1.1.1.1.3"><mtext id="S4.E10.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S4.E10.m1.2.2.1.1.1.1.1.1.3">det</mtext></ci><apply id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1"><minus id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.1"></minus><ci id="S4.E10.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml" 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id="S4.E10.m1.2.2.1.1.3.3.2.3.cmml" xref="S4.E10.m1.2.2.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S4.E10.m1.2.2.1.1.3.3.2.3.1.cmml" xref="S4.E10.m1.2.2.1.1.3.3.2.3">superscript</csymbol><ci id="S4.E10.m1.2.2.1.1.3.3.2.3.2.cmml" xref="S4.E10.m1.2.2.1.1.3.3.2.3.2">𝑧</ci><ci id="S4.E10.m1.2.2.1.1.3.3.2.3.3.cmml" xref="S4.E10.m1.2.2.1.1.3.3.2.3.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10.m1.2c">\left.\text{det}\left(s-\mathcal{A}\right)\right|_{N}=1-\sum_{n=1}^{N}Q_{n}z^{% n}.</annotation><annotation encoding="application/x-llamapun" id="S4.E10.m1.2d">det ( italic_s - caligraphic_A ) | start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 1 - ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_z 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">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p4.4">The coefficients <math alttext="Q_{n}" class="ltx_Math" display="inline" id="S4.p4.3.m1.1"><semantics id="S4.p4.3.m1.1a"><msub id="S4.p4.3.m1.1.1" xref="S4.p4.3.m1.1.1.cmml"><mi id="S4.p4.3.m1.1.1.2" xref="S4.p4.3.m1.1.1.2.cmml">Q</mi><mi id="S4.p4.3.m1.1.1.3" xref="S4.p4.3.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p4.3.m1.1b"><apply id="S4.p4.3.m1.1.1.cmml" xref="S4.p4.3.m1.1.1"><csymbol cd="ambiguous" id="S4.p4.3.m1.1.1.1.cmml" xref="S4.p4.3.m1.1.1">subscript</csymbol><ci id="S4.p4.3.m1.1.1.2.cmml" xref="S4.p4.3.m1.1.1.2">𝑄</ci><ci id="S4.p4.3.m1.1.1.3.cmml" xref="S4.p4.3.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.3.m1.1c">Q_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.3.m1.1d">italic_Q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> are directly related to the <em class="ltx_emph ltx_font_italic" id="S4.p4.4.1">trace coefficients</em> <math alttext="C_{n}=\text{tr}\left(\mathcal{A}^{n}\right)" class="ltx_Math" display="inline" id="S4.p4.4.m2.1"><semantics id="S4.p4.4.m2.1a"><mrow id="S4.p4.4.m2.1.1" xref="S4.p4.4.m2.1.1.cmml"><msub id="S4.p4.4.m2.1.1.3" xref="S4.p4.4.m2.1.1.3.cmml"><mi id="S4.p4.4.m2.1.1.3.2" xref="S4.p4.4.m2.1.1.3.2.cmml">C</mi><mi id="S4.p4.4.m2.1.1.3.3" xref="S4.p4.4.m2.1.1.3.3.cmml">n</mi></msub><mo id="S4.p4.4.m2.1.1.2" xref="S4.p4.4.m2.1.1.2.cmml">=</mo><mrow id="S4.p4.4.m2.1.1.1" xref="S4.p4.4.m2.1.1.1.cmml"><mtext id="S4.p4.4.m2.1.1.1.3" xref="S4.p4.4.m2.1.1.1.3a.cmml">tr</mtext><mo id="S4.p4.4.m2.1.1.1.2" xref="S4.p4.4.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.p4.4.m2.1.1.1.1.1" xref="S4.p4.4.m2.1.1.1.1.1.1.cmml"><mo id="S4.p4.4.m2.1.1.1.1.1.2" 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id="S4.p4.4.m2.1.1.1.3a.cmml" xref="S4.p4.4.m2.1.1.1.3"><mtext id="S4.p4.4.m2.1.1.1.3.cmml" xref="S4.p4.4.m2.1.1.1.3">tr</mtext></ci><apply id="S4.p4.4.m2.1.1.1.1.1.1.cmml" xref="S4.p4.4.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p4.4.m2.1.1.1.1.1.1.1.cmml" xref="S4.p4.4.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.p4.4.m2.1.1.1.1.1.1.2.cmml" xref="S4.p4.4.m2.1.1.1.1.1.1.2">𝒜</ci><ci id="S4.p4.4.m2.1.1.1.1.1.1.3.cmml" xref="S4.p4.4.m2.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.4.m2.1c">C_{n}=\text{tr}\left(\mathcal{A}^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.p4.4.m2.1d">italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = tr ( caligraphic_A start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math> by</p> <table class="ltx_equation ltx_eqn_table" id="S4.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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xref="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.2">𝐶</ci><apply id="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3"><minus id="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3.1.cmml" xref="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3.1"></minus><ci id="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3.2.cmml" xref="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3.2">𝑛</ci><ci id="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3.3.cmml" xref="S4.E11.m1.1.1.1.1.1.1.1.1.3.2.3.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E11.m1.1c">Q_{n}=\frac{1}{n}\left[C_{n}-\sum_{i=1}^{n-1}Q_{i}C_{n-i}\right].</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m1.1d">italic_Q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_n end_ARG [ italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_n - 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">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p4.6">Note that <math alttext="Q_{1}=C_{1}." class="ltx_Math" display="inline" id="S4.p4.5.m1.1"><semantics id="S4.p4.5.m1.1a"><mrow id="S4.p4.5.m1.1.1.1" xref="S4.p4.5.m1.1.1.1.1.cmml"><mrow id="S4.p4.5.m1.1.1.1.1" xref="S4.p4.5.m1.1.1.1.1.cmml"><msub id="S4.p4.5.m1.1.1.1.1.2" xref="S4.p4.5.m1.1.1.1.1.2.cmml"><mi id="S4.p4.5.m1.1.1.1.1.2.2" xref="S4.p4.5.m1.1.1.1.1.2.2.cmml">Q</mi><mn id="S4.p4.5.m1.1.1.1.1.2.3" xref="S4.p4.5.m1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.p4.5.m1.1.1.1.1.1" xref="S4.p4.5.m1.1.1.1.1.1.cmml">=</mo><msub id="S4.p4.5.m1.1.1.1.1.3" xref="S4.p4.5.m1.1.1.1.1.3.cmml"><mi id="S4.p4.5.m1.1.1.1.1.3.2" xref="S4.p4.5.m1.1.1.1.1.3.2.cmml">C</mi><mn id="S4.p4.5.m1.1.1.1.1.3.3" xref="S4.p4.5.m1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S4.p4.5.m1.1.1.1.2" lspace="0em" xref="S4.p4.5.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.5.m1.1b"><apply id="S4.p4.5.m1.1.1.1.1.cmml" xref="S4.p4.5.m1.1.1.1"><eq id="S4.p4.5.m1.1.1.1.1.1.cmml" xref="S4.p4.5.m1.1.1.1.1.1"></eq><apply id="S4.p4.5.m1.1.1.1.1.2.cmml" xref="S4.p4.5.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.p4.5.m1.1.1.1.1.2.1.cmml" xref="S4.p4.5.m1.1.1.1.1.2">subscript</csymbol><ci id="S4.p4.5.m1.1.1.1.1.2.2.cmml" xref="S4.p4.5.m1.1.1.1.1.2.2">𝑄</ci><cn id="S4.p4.5.m1.1.1.1.1.2.3.cmml" type="integer" xref="S4.p4.5.m1.1.1.1.1.2.3">1</cn></apply><apply id="S4.p4.5.m1.1.1.1.1.3.cmml" xref="S4.p4.5.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.p4.5.m1.1.1.1.1.3.1.cmml" xref="S4.p4.5.m1.1.1.1.1.3">subscript</csymbol><ci id="S4.p4.5.m1.1.1.1.1.3.2.cmml" xref="S4.p4.5.m1.1.1.1.1.3.2">𝐶</ci><cn id="S4.p4.5.m1.1.1.1.1.3.3.cmml" type="integer" xref="S4.p4.5.m1.1.1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.5.m1.1c">Q_{1}=C_{1}.</annotation><annotation encoding="application/x-llamapun" id="S4.p4.5.m1.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .</annotation></semantics></math> Finally, the trace coefficients <math alttext="C_{n}" class="ltx_Math" display="inline" id="S4.p4.6.m2.1"><semantics id="S4.p4.6.m2.1a"><msub id="S4.p4.6.m2.1.1" xref="S4.p4.6.m2.1.1.cmml"><mi id="S4.p4.6.m2.1.1.2" xref="S4.p4.6.m2.1.1.2.cmml">C</mi><mi id="S4.p4.6.m2.1.1.3" xref="S4.p4.6.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p4.6.m2.1b"><apply id="S4.p4.6.m2.1.1.cmml" xref="S4.p4.6.m2.1.1"><csymbol cd="ambiguous" id="S4.p4.6.m2.1.1.1.cmml" xref="S4.p4.6.m2.1.1">subscript</csymbol><ci id="S4.p4.6.m2.1.1.2.cmml" xref="S4.p4.6.m2.1.1.2">𝐶</ci><ci id="S4.p4.6.m2.1.1.3.cmml" xref="S4.p4.6.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.6.m2.1c">C_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.6.m2.1d">italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> can be iteratively computed as follows</p> <table class="ltx_equation ltx_eqn_table" id="S4.E12"> <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="\sum_{n=1}^{N}C_{n}z^{n}=\sum_{p}T_{p}\sum_{r=1}^{n_{p}r\leq N}t_{p}^{r}\delta% _{n_{p}r,N}" class="ltx_Math" display="block" id="S4.E12.m1.2"><semantics id="S4.E12.m1.2a"><mrow id="S4.E12.m1.2.3" xref="S4.E12.m1.2.3.cmml"><mrow id="S4.E12.m1.2.3.2" xref="S4.E12.m1.2.3.2.cmml"><munderover id="S4.E12.m1.2.3.2.1" xref="S4.E12.m1.2.3.2.1.cmml"><mo id="S4.E12.m1.2.3.2.1.2.2" movablelimits="false" xref="S4.E12.m1.2.3.2.1.2.2.cmml">∑</mo><mrow id="S4.E12.m1.2.3.2.1.2.3" xref="S4.E12.m1.2.3.2.1.2.3.cmml"><mi id="S4.E12.m1.2.3.2.1.2.3.2" xref="S4.E12.m1.2.3.2.1.2.3.2.cmml">n</mi><mo id="S4.E12.m1.2.3.2.1.2.3.1" xref="S4.E12.m1.2.3.2.1.2.3.1.cmml">=</mo><mn id="S4.E12.m1.2.3.2.1.2.3.3" xref="S4.E12.m1.2.3.2.1.2.3.3.cmml">1</mn></mrow><mi id="S4.E12.m1.2.3.2.1.3" xref="S4.E12.m1.2.3.2.1.3.cmml">N</mi></munderover><mrow id="S4.E12.m1.2.3.2.2" xref="S4.E12.m1.2.3.2.2.cmml"><msub id="S4.E12.m1.2.3.2.2.2" xref="S4.E12.m1.2.3.2.2.2.cmml"><mi id="S4.E12.m1.2.3.2.2.2.2" xref="S4.E12.m1.2.3.2.2.2.2.cmml">C</mi><mi id="S4.E12.m1.2.3.2.2.2.3" xref="S4.E12.m1.2.3.2.2.2.3.cmml">n</mi></msub><mo id="S4.E12.m1.2.3.2.2.1" xref="S4.E12.m1.2.3.2.2.1.cmml">⁢</mo><msup id="S4.E12.m1.2.3.2.2.3" xref="S4.E12.m1.2.3.2.2.3.cmml"><mi id="S4.E12.m1.2.3.2.2.3.2" xref="S4.E12.m1.2.3.2.2.3.2.cmml">z</mi><mi id="S4.E12.m1.2.3.2.2.3.3" xref="S4.E12.m1.2.3.2.2.3.3.cmml">n</mi></msup></mrow></mrow><mo id="S4.E12.m1.2.3.1" rspace="0.111em" xref="S4.E12.m1.2.3.1.cmml">=</mo><mrow id="S4.E12.m1.2.3.3" xref="S4.E12.m1.2.3.3.cmml"><munder id="S4.E12.m1.2.3.3.1" xref="S4.E12.m1.2.3.3.1.cmml"><mo id="S4.E12.m1.2.3.3.1.2" movablelimits="false" xref="S4.E12.m1.2.3.3.1.2.cmml">∑</mo><mi id="S4.E12.m1.2.3.3.1.3" xref="S4.E12.m1.2.3.3.1.3.cmml">p</mi></munder><mrow id="S4.E12.m1.2.3.3.2" xref="S4.E12.m1.2.3.3.2.cmml"><msub id="S4.E12.m1.2.3.3.2.2" xref="S4.E12.m1.2.3.3.2.2.cmml"><mi id="S4.E12.m1.2.3.3.2.2.2" xref="S4.E12.m1.2.3.3.2.2.2.cmml">T</mi><mi id="S4.E12.m1.2.3.3.2.2.3" xref="S4.E12.m1.2.3.3.2.2.3.cmml">p</mi></msub><mo id="S4.E12.m1.2.3.3.2.1" xref="S4.E12.m1.2.3.3.2.1.cmml">⁢</mo><mrow id="S4.E12.m1.2.3.3.2.3" xref="S4.E12.m1.2.3.3.2.3.cmml"><munderover id="S4.E12.m1.2.3.3.2.3.1" xref="S4.E12.m1.2.3.3.2.3.1.cmml"><mo id="S4.E12.m1.2.3.3.2.3.1.2.2" movablelimits="false" xref="S4.E12.m1.2.3.3.2.3.1.2.2.cmml">∑</mo><mrow id="S4.E12.m1.2.3.3.2.3.1.2.3" xref="S4.E12.m1.2.3.3.2.3.1.2.3.cmml"><mi id="S4.E12.m1.2.3.3.2.3.1.2.3.2" xref="S4.E12.m1.2.3.3.2.3.1.2.3.2.cmml">r</mi><mo id="S4.E12.m1.2.3.3.2.3.1.2.3.1" xref="S4.E12.m1.2.3.3.2.3.1.2.3.1.cmml">=</mo><mn id="S4.E12.m1.2.3.3.2.3.1.2.3.3" xref="S4.E12.m1.2.3.3.2.3.1.2.3.3.cmml">1</mn></mrow><mrow id="S4.E12.m1.2.3.3.2.3.1.3" xref="S4.E12.m1.2.3.3.2.3.1.3.cmml"><mrow id="S4.E12.m1.2.3.3.2.3.1.3.2" xref="S4.E12.m1.2.3.3.2.3.1.3.2.cmml"><msub id="S4.E12.m1.2.3.3.2.3.1.3.2.2" xref="S4.E12.m1.2.3.3.2.3.1.3.2.2.cmml"><mi id="S4.E12.m1.2.3.3.2.3.1.3.2.2.2" xref="S4.E12.m1.2.3.3.2.3.1.3.2.2.2.cmml">n</mi><mi id="S4.E12.m1.2.3.3.2.3.1.3.2.2.3" xref="S4.E12.m1.2.3.3.2.3.1.3.2.2.3.cmml">p</mi></msub><mo id="S4.E12.m1.2.3.3.2.3.1.3.2.1" xref="S4.E12.m1.2.3.3.2.3.1.3.2.1.cmml">⁢</mo><mi id="S4.E12.m1.2.3.3.2.3.1.3.2.3" xref="S4.E12.m1.2.3.3.2.3.1.3.2.3.cmml">r</mi></mrow><mo 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xref="S4.E12.m1.2.3.3.2.3.2.1"></times><apply id="S4.E12.m1.2.3.3.2.3.2.2.cmml" xref="S4.E12.m1.2.3.3.2.3.2.2"><csymbol cd="ambiguous" id="S4.E12.m1.2.3.3.2.3.2.2.1.cmml" xref="S4.E12.m1.2.3.3.2.3.2.2">superscript</csymbol><apply id="S4.E12.m1.2.3.3.2.3.2.2.2.cmml" xref="S4.E12.m1.2.3.3.2.3.2.2"><csymbol cd="ambiguous" id="S4.E12.m1.2.3.3.2.3.2.2.2.1.cmml" xref="S4.E12.m1.2.3.3.2.3.2.2">subscript</csymbol><ci id="S4.E12.m1.2.3.3.2.3.2.2.2.2.cmml" xref="S4.E12.m1.2.3.3.2.3.2.2.2.2">𝑡</ci><ci id="S4.E12.m1.2.3.3.2.3.2.2.2.3.cmml" xref="S4.E12.m1.2.3.3.2.3.2.2.2.3">𝑝</ci></apply><ci id="S4.E12.m1.2.3.3.2.3.2.2.3.cmml" xref="S4.E12.m1.2.3.3.2.3.2.2.3">𝑟</ci></apply><apply id="S4.E12.m1.2.3.3.2.3.2.3.cmml" xref="S4.E12.m1.2.3.3.2.3.2.3"><csymbol cd="ambiguous" id="S4.E12.m1.2.3.3.2.3.2.3.1.cmml" xref="S4.E12.m1.2.3.3.2.3.2.3">subscript</csymbol><ci id="S4.E12.m1.2.3.3.2.3.2.3.2.cmml" xref="S4.E12.m1.2.3.3.2.3.2.3.2">𝛿</ci><list id="S4.E12.m1.2.2.2.3.cmml" xref="S4.E12.m1.2.2.2.2"><apply id="S4.E12.m1.2.2.2.2.1.cmml" xref="S4.E12.m1.2.2.2.2.1"><times id="S4.E12.m1.2.2.2.2.1.1.cmml" xref="S4.E12.m1.2.2.2.2.1.1"></times><apply id="S4.E12.m1.2.2.2.2.1.2.cmml" xref="S4.E12.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.E12.m1.2.2.2.2.1.2.1.cmml" xref="S4.E12.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S4.E12.m1.2.2.2.2.1.2.2.cmml" xref="S4.E12.m1.2.2.2.2.1.2.2">𝑛</ci><ci id="S4.E12.m1.2.2.2.2.1.2.3.cmml" xref="S4.E12.m1.2.2.2.2.1.2.3">𝑝</ci></apply><ci id="S4.E12.m1.2.2.2.2.1.3.cmml" xref="S4.E12.m1.2.2.2.2.1.3">𝑟</ci></apply><ci id="S4.E12.m1.1.1.1.1.cmml" xref="S4.E12.m1.1.1.1.1">𝑁</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E12.m1.2c">\sum_{n=1}^{N}C_{n}z^{n}=\sum_{p}T_{p}\sum_{r=1}^{n_{p}r\leq N}t_{p}^{r}\delta% _{n_{p}r,N}</annotation><annotation encoding="application/x-llamapun" id="S4.E12.m1.2d">∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_r = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_r ≤ italic_N end_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_r , italic_N 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">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p4.16">where,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E13"> <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="t_{p}=\frac{z^{n_{p}}e^{-sT_{p}}}{\left|\Lambda_{p}\right|}." class="ltx_Math" display="block" id="S4.E13.m1.2"><semantics id="S4.E13.m1.2a"><mrow id="S4.E13.m1.2.2.1" xref="S4.E13.m1.2.2.1.1.cmml"><mrow id="S4.E13.m1.2.2.1.1" xref="S4.E13.m1.2.2.1.1.cmml"><msub id="S4.E13.m1.2.2.1.1.2" xref="S4.E13.m1.2.2.1.1.2.cmml"><mi id="S4.E13.m1.2.2.1.1.2.2" xref="S4.E13.m1.2.2.1.1.2.2.cmml">t</mi><mi id="S4.E13.m1.2.2.1.1.2.3" xref="S4.E13.m1.2.2.1.1.2.3.cmml">p</mi></msub><mo id="S4.E13.m1.2.2.1.1.1" xref="S4.E13.m1.2.2.1.1.1.cmml">=</mo><mfrac id="S4.E13.m1.1.1" xref="S4.E13.m1.1.1.cmml"><mrow id="S4.E13.m1.1.1.3" xref="S4.E13.m1.1.1.3.cmml"><msup id="S4.E13.m1.1.1.3.2" xref="S4.E13.m1.1.1.3.2.cmml"><mi id="S4.E13.m1.1.1.3.2.2" xref="S4.E13.m1.1.1.3.2.2.cmml">z</mi><msub id="S4.E13.m1.1.1.3.2.3" xref="S4.E13.m1.1.1.3.2.3.cmml"><mi id="S4.E13.m1.1.1.3.2.3.2" xref="S4.E13.m1.1.1.3.2.3.2.cmml">n</mi><mi id="S4.E13.m1.1.1.3.2.3.3" xref="S4.E13.m1.1.1.3.2.3.3.cmml">p</mi></msub></msup><mo id="S4.E13.m1.1.1.3.1" xref="S4.E13.m1.1.1.3.1.cmml">⁢</mo><msup id="S4.E13.m1.1.1.3.3" xref="S4.E13.m1.1.1.3.3.cmml"><mi id="S4.E13.m1.1.1.3.3.2" xref="S4.E13.m1.1.1.3.3.2.cmml">e</mi><mrow id="S4.E13.m1.1.1.3.3.3" xref="S4.E13.m1.1.1.3.3.3.cmml"><mo id="S4.E13.m1.1.1.3.3.3a" xref="S4.E13.m1.1.1.3.3.3.cmml">−</mo><mrow id="S4.E13.m1.1.1.3.3.3.2" xref="S4.E13.m1.1.1.3.3.3.2.cmml"><mi id="S4.E13.m1.1.1.3.3.3.2.2" xref="S4.E13.m1.1.1.3.3.3.2.2.cmml">s</mi><mo id="S4.E13.m1.1.1.3.3.3.2.1" xref="S4.E13.m1.1.1.3.3.3.2.1.cmml">⁢</mo><msub id="S4.E13.m1.1.1.3.3.3.2.3" xref="S4.E13.m1.1.1.3.3.3.2.3.cmml"><mi id="S4.E13.m1.1.1.3.3.3.2.3.2" xref="S4.E13.m1.1.1.3.3.3.2.3.2.cmml">T</mi><mi id="S4.E13.m1.1.1.3.3.3.2.3.3" xref="S4.E13.m1.1.1.3.3.3.2.3.3.cmml">p</mi></msub></mrow></mrow></msup></mrow><mrow id="S4.E13.m1.1.1.1.1" xref="S4.E13.m1.1.1.1.2.cmml"><mo id="S4.E13.m1.1.1.1.1.2" xref="S4.E13.m1.1.1.1.2.1.cmml">|</mo><msub id="S4.E13.m1.1.1.1.1.1" xref="S4.E13.m1.1.1.1.1.1.cmml"><mi id="S4.E13.m1.1.1.1.1.1.2" mathvariant="normal" xref="S4.E13.m1.1.1.1.1.1.2.cmml">Λ</mi><mi id="S4.E13.m1.1.1.1.1.1.3" xref="S4.E13.m1.1.1.1.1.1.3.cmml">p</mi></msub><mo id="S4.E13.m1.1.1.1.1.3" xref="S4.E13.m1.1.1.1.2.1.cmml">|</mo></mrow></mfrac></mrow><mo id="S4.E13.m1.2.2.1.2" lspace="0em" xref="S4.E13.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E13.m1.2b"><apply id="S4.E13.m1.2.2.1.1.cmml" xref="S4.E13.m1.2.2.1"><eq id="S4.E13.m1.2.2.1.1.1.cmml" xref="S4.E13.m1.2.2.1.1.1"></eq><apply id="S4.E13.m1.2.2.1.1.2.cmml" xref="S4.E13.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="S4.E13.m1.2.2.1.1.2.1.cmml" xref="S4.E13.m1.2.2.1.1.2">subscript</csymbol><ci id="S4.E13.m1.2.2.1.1.2.2.cmml" xref="S4.E13.m1.2.2.1.1.2.2">𝑡</ci><ci id="S4.E13.m1.2.2.1.1.2.3.cmml" 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id="S4.E13.m1.1.1.3.3.2.cmml" xref="S4.E13.m1.1.1.3.3.2">𝑒</ci><apply id="S4.E13.m1.1.1.3.3.3.cmml" xref="S4.E13.m1.1.1.3.3.3"><minus id="S4.E13.m1.1.1.3.3.3.1.cmml" xref="S4.E13.m1.1.1.3.3.3"></minus><apply id="S4.E13.m1.1.1.3.3.3.2.cmml" xref="S4.E13.m1.1.1.3.3.3.2"><times id="S4.E13.m1.1.1.3.3.3.2.1.cmml" xref="S4.E13.m1.1.1.3.3.3.2.1"></times><ci id="S4.E13.m1.1.1.3.3.3.2.2.cmml" xref="S4.E13.m1.1.1.3.3.3.2.2">𝑠</ci><apply id="S4.E13.m1.1.1.3.3.3.2.3.cmml" xref="S4.E13.m1.1.1.3.3.3.2.3"><csymbol cd="ambiguous" id="S4.E13.m1.1.1.3.3.3.2.3.1.cmml" xref="S4.E13.m1.1.1.3.3.3.2.3">subscript</csymbol><ci id="S4.E13.m1.1.1.3.3.3.2.3.2.cmml" xref="S4.E13.m1.1.1.3.3.3.2.3.2">𝑇</ci><ci id="S4.E13.m1.1.1.3.3.3.2.3.3.cmml" xref="S4.E13.m1.1.1.3.3.3.2.3.3">𝑝</ci></apply></apply></apply></apply></apply><apply id="S4.E13.m1.1.1.1.2.cmml" xref="S4.E13.m1.1.1.1.1"><abs id="S4.E13.m1.1.1.1.2.1.cmml" xref="S4.E13.m1.1.1.1.1.2"></abs><apply id="S4.E13.m1.1.1.1.1.1.cmml" xref="S4.E13.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E13.m1.1.1.1.1.1.1.cmml" xref="S4.E13.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.E13.m1.1.1.1.1.1.2.cmml" xref="S4.E13.m1.1.1.1.1.1.2">Λ</ci><ci id="S4.E13.m1.1.1.1.1.1.3.cmml" xref="S4.E13.m1.1.1.1.1.1.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13.m1.2c">t_{p}=\frac{z^{n_{p}}e^{-sT_{p}}}{\left|\Lambda_{p}\right|}.</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m1.2d">italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = divide start_ARG italic_z start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_s italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG | roman_Λ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT | 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">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p4.15">Here <math alttext="t_{p}" class="ltx_Math" display="inline" id="S4.p4.7.m1.1"><semantics id="S4.p4.7.m1.1a"><msub id="S4.p4.7.m1.1.1" xref="S4.p4.7.m1.1.1.cmml"><mi id="S4.p4.7.m1.1.1.2" xref="S4.p4.7.m1.1.1.2.cmml">t</mi><mi id="S4.p4.7.m1.1.1.3" xref="S4.p4.7.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p4.7.m1.1b"><apply id="S4.p4.7.m1.1.1.cmml" xref="S4.p4.7.m1.1.1"><csymbol cd="ambiguous" id="S4.p4.7.m1.1.1.1.cmml" xref="S4.p4.7.m1.1.1">subscript</csymbol><ci id="S4.p4.7.m1.1.1.2.cmml" xref="S4.p4.7.m1.1.1.2">𝑡</ci><ci id="S4.p4.7.m1.1.1.3.cmml" xref="S4.p4.7.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.7.m1.1c">t_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.7.m1.1d">italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is the <span class="ltx_text ltx_inline-quote ltx_outerquote" id="S4.p4.15.1">“local trace”</span> associated with each <math alttext="p" class="ltx_Math" display="inline" id="S4.p4.8.m2.1"><semantics id="S4.p4.8.m2.1a"><mi id="S4.p4.8.m2.1.1" xref="S4.p4.8.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.p4.8.m2.1b"><ci id="S4.p4.8.m2.1.1.cmml" xref="S4.p4.8.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.8.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.p4.8.m2.1d">italic_p</annotation></semantics></math> cycle that acts as the weight for each periodic orbit and <math alttext="n_{p}" class="ltx_Math" display="inline" id="S4.p4.9.m3.1"><semantics id="S4.p4.9.m3.1a"><msub id="S4.p4.9.m3.1.1" xref="S4.p4.9.m3.1.1.cmml"><mi id="S4.p4.9.m3.1.1.2" xref="S4.p4.9.m3.1.1.2.cmml">n</mi><mi id="S4.p4.9.m3.1.1.3" xref="S4.p4.9.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p4.9.m3.1b"><apply id="S4.p4.9.m3.1.1.cmml" xref="S4.p4.9.m3.1.1"><csymbol cd="ambiguous" id="S4.p4.9.m3.1.1.1.cmml" xref="S4.p4.9.m3.1.1">subscript</csymbol><ci id="S4.p4.9.m3.1.1.2.cmml" xref="S4.p4.9.m3.1.1.2">𝑛</ci><ci id="S4.p4.9.m3.1.1.3.cmml" xref="S4.p4.9.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.9.m3.1c">n_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.9.m3.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is the discrete-time period. For continuous dynamics we set <math alttext="z=1" class="ltx_Math" display="inline" id="S4.p4.10.m4.1"><semantics id="S4.p4.10.m4.1a"><mrow id="S4.p4.10.m4.1.1" xref="S4.p4.10.m4.1.1.cmml"><mi id="S4.p4.10.m4.1.1.2" xref="S4.p4.10.m4.1.1.2.cmml">z</mi><mo id="S4.p4.10.m4.1.1.1" xref="S4.p4.10.m4.1.1.1.cmml">=</mo><mn id="S4.p4.10.m4.1.1.3" xref="S4.p4.10.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.10.m4.1b"><apply id="S4.p4.10.m4.1.1.cmml" xref="S4.p4.10.m4.1.1"><eq id="S4.p4.10.m4.1.1.1.cmml" xref="S4.p4.10.m4.1.1.1"></eq><ci id="S4.p4.10.m4.1.1.2.cmml" xref="S4.p4.10.m4.1.1.2">𝑧</ci><cn id="S4.p4.10.m4.1.1.3.cmml" type="integer" xref="S4.p4.10.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.10.m4.1c">z=1</annotation><annotation encoding="application/x-llamapun" id="S4.p4.10.m4.1d">italic_z = 1</annotation></semantics></math> and find the zeros of the resulting function of <math alttext="s" class="ltx_Math" display="inline" id="S4.p4.11.m5.1"><semantics id="S4.p4.11.m5.1a"><mi id="S4.p4.11.m5.1.1" xref="S4.p4.11.m5.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S4.p4.11.m5.1b"><ci id="S4.p4.11.m5.1.1.cmml" xref="S4.p4.11.m5.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.11.m5.1c">s</annotation><annotation encoding="application/x-llamapun" id="S4.p4.11.m5.1d">italic_s</annotation></semantics></math>. For discrete dynamics we set <math alttext="s=0" class="ltx_Math" display="inline" id="S4.p4.12.m6.1"><semantics id="S4.p4.12.m6.1a"><mrow id="S4.p4.12.m6.1.1" xref="S4.p4.12.m6.1.1.cmml"><mi id="S4.p4.12.m6.1.1.2" xref="S4.p4.12.m6.1.1.2.cmml">s</mi><mo id="S4.p4.12.m6.1.1.1" xref="S4.p4.12.m6.1.1.1.cmml">=</mo><mn id="S4.p4.12.m6.1.1.3" xref="S4.p4.12.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.12.m6.1b"><apply id="S4.p4.12.m6.1.1.cmml" xref="S4.p4.12.m6.1.1"><eq id="S4.p4.12.m6.1.1.1.cmml" xref="S4.p4.12.m6.1.1.1"></eq><ci id="S4.p4.12.m6.1.1.2.cmml" xref="S4.p4.12.m6.1.1.2">𝑠</ci><cn id="S4.p4.12.m6.1.1.3.cmml" type="integer" xref="S4.p4.12.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.12.m6.1c">s=0</annotation><annotation encoding="application/x-llamapun" id="S4.p4.12.m6.1d">italic_s = 0</annotation></semantics></math> and find the roots of the resulting polynomial in <math alttext="z" class="ltx_Math" display="inline" id="S4.p4.13.m7.1"><semantics id="S4.p4.13.m7.1a"><mi id="S4.p4.13.m7.1.1" xref="S4.p4.13.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.p4.13.m7.1b"><ci id="S4.p4.13.m7.1.1.cmml" xref="S4.p4.13.m7.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.13.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.p4.13.m7.1d">italic_z</annotation></semantics></math>. Note also one must switch from continuous-time period <math alttext="T_{p}" class="ltx_Math" display="inline" id="S4.p4.14.m8.1"><semantics id="S4.p4.14.m8.1a"><msub id="S4.p4.14.m8.1.1" xref="S4.p4.14.m8.1.1.cmml"><mi id="S4.p4.14.m8.1.1.2" xref="S4.p4.14.m8.1.1.2.cmml">T</mi><mi id="S4.p4.14.m8.1.1.3" xref="S4.p4.14.m8.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p4.14.m8.1b"><apply id="S4.p4.14.m8.1.1.cmml" xref="S4.p4.14.m8.1.1"><csymbol cd="ambiguous" id="S4.p4.14.m8.1.1.1.cmml" xref="S4.p4.14.m8.1.1">subscript</csymbol><ci id="S4.p4.14.m8.1.1.2.cmml" xref="S4.p4.14.m8.1.1.2">𝑇</ci><ci id="S4.p4.14.m8.1.1.3.cmml" xref="S4.p4.14.m8.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.14.m8.1c">T_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.14.m8.1d">italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> to discrete-time period <math alttext="n_{p}" class="ltx_Math" display="inline" id="S4.p4.15.m9.1"><semantics id="S4.p4.15.m9.1a"><msub id="S4.p4.15.m9.1.1" xref="S4.p4.15.m9.1.1.cmml"><mi id="S4.p4.15.m9.1.1.2" xref="S4.p4.15.m9.1.1.2.cmml">n</mi><mi id="S4.p4.15.m9.1.1.3" xref="S4.p4.15.m9.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p4.15.m9.1b"><apply id="S4.p4.15.m9.1.1.cmml" xref="S4.p4.15.m9.1.1"><csymbol cd="ambiguous" id="S4.p4.15.m9.1.1.1.cmml" xref="S4.p4.15.m9.1.1">subscript</csymbol><ci id="S4.p4.15.m9.1.1.2.cmml" xref="S4.p4.15.m9.1.1.2">𝑛</ci><ci id="S4.p4.15.m9.1.1.3.cmml" xref="S4.p4.15.m9.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.15.m9.1c">n_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.15.m9.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.2">There are several key advantages to using periodic orbit techniques over Monte Carlo methods. First, orders of magnitude fewer trajectories are needed to compute the escape rate. Monte Carlo simulations require <math alttext="10^{7}" class="ltx_Math" display="inline" id="S4.p5.1.m1.1"><semantics id="S4.p5.1.m1.1a"><msup id="S4.p5.1.m1.1.1" xref="S4.p5.1.m1.1.1.cmml"><mn id="S4.p5.1.m1.1.1.2" xref="S4.p5.1.m1.1.1.2.cmml">10</mn><mn id="S4.p5.1.m1.1.1.3" xref="S4.p5.1.m1.1.1.3.cmml">7</mn></msup><annotation-xml encoding="MathML-Content" id="S4.p5.1.m1.1b"><apply id="S4.p5.1.m1.1.1.cmml" xref="S4.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p5.1.m1.1.1.1.cmml" xref="S4.p5.1.m1.1.1">superscript</csymbol><cn id="S4.p5.1.m1.1.1.2.cmml" type="integer" xref="S4.p5.1.m1.1.1.2">10</cn><cn id="S4.p5.1.m1.1.1.3.cmml" type="integer" xref="S4.p5.1.m1.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.1.m1.1c">10^{7}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.1.m1.1d">10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT</annotation></semantics></math> trajectories while periodic orbits methods require orders of magnitude fewer. In this paper we use only <math alttext="10^{4}" class="ltx_Math" display="inline" id="S4.p5.2.m2.1"><semantics id="S4.p5.2.m2.1a"><msup id="S4.p5.2.m2.1.1" xref="S4.p5.2.m2.1.1.cmml"><mn id="S4.p5.2.m2.1.1.2" xref="S4.p5.2.m2.1.1.2.cmml">10</mn><mn id="S4.p5.2.m2.1.1.3" xref="S4.p5.2.m2.1.1.3.cmml">4</mn></msup><annotation-xml encoding="MathML-Content" id="S4.p5.2.m2.1b"><apply id="S4.p5.2.m2.1.1.cmml" xref="S4.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p5.2.m2.1.1.1.cmml" xref="S4.p5.2.m2.1.1">superscript</csymbol><cn id="S4.p5.2.m2.1.1.2.cmml" type="integer" xref="S4.p5.2.m2.1.1.2">10</cn><cn id="S4.p5.2.m2.1.1.3.cmml" type="integer" xref="S4.p5.2.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.2.m2.1c">10^{4}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.2.m2.1d">10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math> periodic trajectories. Second, they give a more complete understanding of the underlying dynamics of the system. Decomposing the dynamics into a trace over periodic orbits provides a more robust physical framework to view the system under. Next, other bulk physical properties besides escape rates can be computed with periodic orbits such as entropies, transport coefficients, and even quantum resonances, using a semiclassical theory. Finally, as parameters are varied, periodic orbits change continuously enabling us to numerically continue orbits through parameter space rather than running an entirely new Monte Carlo simulation for each parameter value.</p> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span>Surface of Section and Discrete-Time Monte Carlo</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.13">Within the full <math alttext="uvp_{u}p_{v}" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mrow id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">u</mi><mo id="S5.p1.1.m1.1.1.1" xref="S5.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S5.p1.1.m1.1.1.3" xref="S5.p1.1.m1.1.1.3.cmml">v</mi><mo id="S5.p1.1.m1.1.1.1a" xref="S5.p1.1.m1.1.1.1.cmml">⁢</mo><msub id="S5.p1.1.m1.1.1.4" xref="S5.p1.1.m1.1.1.4.cmml"><mi id="S5.p1.1.m1.1.1.4.2" xref="S5.p1.1.m1.1.1.4.2.cmml">p</mi><mi id="S5.p1.1.m1.1.1.4.3" xref="S5.p1.1.m1.1.1.4.3.cmml">u</mi></msub><mo id="S5.p1.1.m1.1.1.1b" xref="S5.p1.1.m1.1.1.1.cmml">⁢</mo><msub id="S5.p1.1.m1.1.1.5" xref="S5.p1.1.m1.1.1.5.cmml"><mi id="S5.p1.1.m1.1.1.5.2" xref="S5.p1.1.m1.1.1.5.2.cmml">p</mi><mi id="S5.p1.1.m1.1.1.5.3" xref="S5.p1.1.m1.1.1.5.3.cmml">v</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><times id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1.1"></times><ci id="S5.p1.1.m1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.2">𝑢</ci><ci id="S5.p1.1.m1.1.1.3.cmml" xref="S5.p1.1.m1.1.1.3">𝑣</ci><apply id="S5.p1.1.m1.1.1.4.cmml" xref="S5.p1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S5.p1.1.m1.1.1.4.1.cmml" xref="S5.p1.1.m1.1.1.4">subscript</csymbol><ci id="S5.p1.1.m1.1.1.4.2.cmml" xref="S5.p1.1.m1.1.1.4.2">𝑝</ci><ci id="S5.p1.1.m1.1.1.4.3.cmml" xref="S5.p1.1.m1.1.1.4.3">𝑢</ci></apply><apply id="S5.p1.1.m1.1.1.5.cmml" xref="S5.p1.1.m1.1.1.5"><csymbol cd="ambiguous" id="S5.p1.1.m1.1.1.5.1.cmml" xref="S5.p1.1.m1.1.1.5">subscript</csymbol><ci id="S5.p1.1.m1.1.1.5.2.cmml" xref="S5.p1.1.m1.1.1.5.2">𝑝</ci><ci id="S5.p1.1.m1.1.1.5.3.cmml" xref="S5.p1.1.m1.1.1.5.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">uvp_{u}p_{v}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">italic_u italic_v italic_p start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> phase space we define a two-dimensional surface, called the surface of section, via the constraints <math alttext="h=0" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mrow id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><mi id="S5.p1.2.m2.1.1.2" xref="S5.p1.2.m2.1.1.2.cmml">h</mi><mo id="S5.p1.2.m2.1.1.1" xref="S5.p1.2.m2.1.1.1.cmml">=</mo><mn id="S5.p1.2.m2.1.1.3" xref="S5.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"><eq id="S5.p1.2.m2.1.1.1.cmml" xref="S5.p1.2.m2.1.1.1"></eq><ci id="S5.p1.2.m2.1.1.2.cmml" xref="S5.p1.2.m2.1.1.2">ℎ</ci><cn id="S5.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">h=0</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">italic_h = 0</annotation></semantics></math>, <math alttext="u=0" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><mrow id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2" xref="S5.p1.3.m3.1.1.2.cmml">u</mi><mo id="S5.p1.3.m3.1.1.1" xref="S5.p1.3.m3.1.1.1.cmml">=</mo><mn id="S5.p1.3.m3.1.1.3" xref="S5.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><eq id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1.1"></eq><ci id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1.2">𝑢</ci><cn id="S5.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">u=0</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">italic_u = 0</annotation></semantics></math>. This surface can be parameterized by <math alttext="\left(v,p_{v}\right)" class="ltx_Math" display="inline" id="S5.p1.4.m4.2"><semantics id="S5.p1.4.m4.2a"><mrow id="S5.p1.4.m4.2.2.1" xref="S5.p1.4.m4.2.2.2.cmml"><mo id="S5.p1.4.m4.2.2.1.2" xref="S5.p1.4.m4.2.2.2.cmml">(</mo><mi id="S5.p1.4.m4.1.1" xref="S5.p1.4.m4.1.1.cmml">v</mi><mo id="S5.p1.4.m4.2.2.1.3" xref="S5.p1.4.m4.2.2.2.cmml">,</mo><msub id="S5.p1.4.m4.2.2.1.1" xref="S5.p1.4.m4.2.2.1.1.cmml"><mi id="S5.p1.4.m4.2.2.1.1.2" xref="S5.p1.4.m4.2.2.1.1.2.cmml">p</mi><mi id="S5.p1.4.m4.2.2.1.1.3" xref="S5.p1.4.m4.2.2.1.1.3.cmml">v</mi></msub><mo id="S5.p1.4.m4.2.2.1.4" xref="S5.p1.4.m4.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.4.m4.2b"><interval closure="open" id="S5.p1.4.m4.2.2.2.cmml" xref="S5.p1.4.m4.2.2.1"><ci id="S5.p1.4.m4.1.1.cmml" xref="S5.p1.4.m4.1.1">𝑣</ci><apply id="S5.p1.4.m4.2.2.1.1.cmml" xref="S5.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S5.p1.4.m4.2.2.1.1.1.cmml" xref="S5.p1.4.m4.2.2.1.1">subscript</csymbol><ci id="S5.p1.4.m4.2.2.1.1.2.cmml" xref="S5.p1.4.m4.2.2.1.1.2">𝑝</ci><ci id="S5.p1.4.m4.2.2.1.1.3.cmml" xref="S5.p1.4.m4.2.2.1.1.3">𝑣</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.2c">\left(v,p_{v}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.4.m4.2d">( italic_v , italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT )</annotation></semantics></math>. An initial point in the <math alttext="vp_{v}" class="ltx_Math" display="inline" id="S5.p1.5.m5.1"><semantics id="S5.p1.5.m5.1a"><mrow id="S5.p1.5.m5.1.1" xref="S5.p1.5.m5.1.1.cmml"><mi id="S5.p1.5.m5.1.1.2" xref="S5.p1.5.m5.1.1.2.cmml">v</mi><mo id="S5.p1.5.m5.1.1.1" xref="S5.p1.5.m5.1.1.1.cmml">⁢</mo><msub id="S5.p1.5.m5.1.1.3" xref="S5.p1.5.m5.1.1.3.cmml"><mi id="S5.p1.5.m5.1.1.3.2" xref="S5.p1.5.m5.1.1.3.2.cmml">p</mi><mi id="S5.p1.5.m5.1.1.3.3" xref="S5.p1.5.m5.1.1.3.3.cmml">v</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.5.m5.1b"><apply id="S5.p1.5.m5.1.1.cmml" xref="S5.p1.5.m5.1.1"><times id="S5.p1.5.m5.1.1.1.cmml" xref="S5.p1.5.m5.1.1.1"></times><ci id="S5.p1.5.m5.1.1.2.cmml" xref="S5.p1.5.m5.1.1.2">𝑣</ci><apply id="S5.p1.5.m5.1.1.3.cmml" xref="S5.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.p1.5.m5.1.1.3.1.cmml" xref="S5.p1.5.m5.1.1.3">subscript</csymbol><ci id="S5.p1.5.m5.1.1.3.2.cmml" xref="S5.p1.5.m5.1.1.3.2">𝑝</ci><ci id="S5.p1.5.m5.1.1.3.3.cmml" xref="S5.p1.5.m5.1.1.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.5.m5.1c">vp_{v}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.5.m5.1d">italic_v italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> plane is then mapped forward according to <math alttext="\mathcal{M}\left(v_{0},\,p_{v0}\right)=\left(v_{1},\,p_{v1}\right)" class="ltx_Math" display="inline" id="S5.p1.6.m6.4"><semantics id="S5.p1.6.m6.4a"><mrow id="S5.p1.6.m6.4.4" xref="S5.p1.6.m6.4.4.cmml"><mrow id="S5.p1.6.m6.2.2.2" xref="S5.p1.6.m6.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p1.6.m6.2.2.2.4" xref="S5.p1.6.m6.2.2.2.4.cmml">ℳ</mi><mo id="S5.p1.6.m6.2.2.2.3" xref="S5.p1.6.m6.2.2.2.3.cmml">⁢</mo><mrow id="S5.p1.6.m6.2.2.2.2.2" xref="S5.p1.6.m6.2.2.2.2.3.cmml"><mo id="S5.p1.6.m6.2.2.2.2.2.3" xref="S5.p1.6.m6.2.2.2.2.3.cmml">(</mo><msub id="S5.p1.6.m6.1.1.1.1.1.1" xref="S5.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.p1.6.m6.1.1.1.1.1.1.2" xref="S5.p1.6.m6.1.1.1.1.1.1.2.cmml">v</mi><mn id="S5.p1.6.m6.1.1.1.1.1.1.3" xref="S5.p1.6.m6.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S5.p1.6.m6.2.2.2.2.2.4" rspace="0.337em" xref="S5.p1.6.m6.2.2.2.2.3.cmml">,</mo><msub id="S5.p1.6.m6.2.2.2.2.2.2" xref="S5.p1.6.m6.2.2.2.2.2.2.cmml"><mi id="S5.p1.6.m6.2.2.2.2.2.2.2" xref="S5.p1.6.m6.2.2.2.2.2.2.2.cmml">p</mi><mrow id="S5.p1.6.m6.2.2.2.2.2.2.3" xref="S5.p1.6.m6.2.2.2.2.2.2.3.cmml"><mi id="S5.p1.6.m6.2.2.2.2.2.2.3.2" xref="S5.p1.6.m6.2.2.2.2.2.2.3.2.cmml">v</mi><mo id="S5.p1.6.m6.2.2.2.2.2.2.3.1" xref="S5.p1.6.m6.2.2.2.2.2.2.3.1.cmml">⁢</mo><mn id="S5.p1.6.m6.2.2.2.2.2.2.3.3" xref="S5.p1.6.m6.2.2.2.2.2.2.3.3.cmml">0</mn></mrow></msub><mo id="S5.p1.6.m6.2.2.2.2.2.5" xref="S5.p1.6.m6.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.p1.6.m6.4.4.5" xref="S5.p1.6.m6.4.4.5.cmml">=</mo><mrow id="S5.p1.6.m6.4.4.4.2" xref="S5.p1.6.m6.4.4.4.3.cmml"><mo id="S5.p1.6.m6.4.4.4.2.3" xref="S5.p1.6.m6.4.4.4.3.cmml">(</mo><msub 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xref="S5.p1.6.m6.4.4.5"></eq><apply id="S5.p1.6.m6.2.2.2.cmml" xref="S5.p1.6.m6.2.2.2"><times id="S5.p1.6.m6.2.2.2.3.cmml" xref="S5.p1.6.m6.2.2.2.3"></times><ci id="S5.p1.6.m6.2.2.2.4.cmml" xref="S5.p1.6.m6.2.2.2.4">ℳ</ci><interval closure="open" id="S5.p1.6.m6.2.2.2.2.3.cmml" xref="S5.p1.6.m6.2.2.2.2.2"><apply id="S5.p1.6.m6.1.1.1.1.1.1.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1.2">𝑣</ci><cn id="S5.p1.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.p1.6.m6.1.1.1.1.1.1.3">0</cn></apply><apply id="S5.p1.6.m6.2.2.2.2.2.2.cmml" xref="S5.p1.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.p1.6.m6.2.2.2.2.2.2.1.cmml" xref="S5.p1.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S5.p1.6.m6.2.2.2.2.2.2.2.cmml" xref="S5.p1.6.m6.2.2.2.2.2.2.2">𝑝</ci><apply id="S5.p1.6.m6.2.2.2.2.2.2.3.cmml" xref="S5.p1.6.m6.2.2.2.2.2.2.3"><times id="S5.p1.6.m6.2.2.2.2.2.2.3.1.cmml" xref="S5.p1.6.m6.2.2.2.2.2.2.3.1"></times><ci id="S5.p1.6.m6.2.2.2.2.2.2.3.2.cmml" xref="S5.p1.6.m6.2.2.2.2.2.2.3.2">𝑣</ci><cn id="S5.p1.6.m6.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S5.p1.6.m6.2.2.2.2.2.2.3.3">0</cn></apply></apply></interval></apply><interval closure="open" id="S5.p1.6.m6.4.4.4.3.cmml" xref="S5.p1.6.m6.4.4.4.2"><apply id="S5.p1.6.m6.3.3.3.1.1.cmml" xref="S5.p1.6.m6.3.3.3.1.1"><csymbol cd="ambiguous" id="S5.p1.6.m6.3.3.3.1.1.1.cmml" xref="S5.p1.6.m6.3.3.3.1.1">subscript</csymbol><ci id="S5.p1.6.m6.3.3.3.1.1.2.cmml" xref="S5.p1.6.m6.3.3.3.1.1.2">𝑣</ci><cn id="S5.p1.6.m6.3.3.3.1.1.3.cmml" type="integer" xref="S5.p1.6.m6.3.3.3.1.1.3">1</cn></apply><apply id="S5.p1.6.m6.4.4.4.2.2.cmml" xref="S5.p1.6.m6.4.4.4.2.2"><csymbol cd="ambiguous" id="S5.p1.6.m6.4.4.4.2.2.1.cmml" xref="S5.p1.6.m6.4.4.4.2.2">subscript</csymbol><ci id="S5.p1.6.m6.4.4.4.2.2.2.cmml" xref="S5.p1.6.m6.4.4.4.2.2.2">𝑝</ci><apply id="S5.p1.6.m6.4.4.4.2.2.3.cmml" xref="S5.p1.6.m6.4.4.4.2.2.3"><times id="S5.p1.6.m6.4.4.4.2.2.3.1.cmml" xref="S5.p1.6.m6.4.4.4.2.2.3.1"></times><ci id="S5.p1.6.m6.4.4.4.2.2.3.2.cmml" xref="S5.p1.6.m6.4.4.4.2.2.3.2">𝑣</ci><cn id="S5.p1.6.m6.4.4.4.2.2.3.3.cmml" type="integer" xref="S5.p1.6.m6.4.4.4.2.2.3.3">1</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.6.m6.4c">\mathcal{M}\left(v_{0},\,p_{v0}\right)=\left(v_{1},\,p_{v1}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.6.m6.4d">caligraphic_M ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_v 0 end_POSTSUBSCRIPT ) = ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_v 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Numerically, this map takes a point <math alttext="\left(v_{0},\,p_{v0}\right)" class="ltx_Math" display="inline" id="S5.p1.7.m7.2"><semantics id="S5.p1.7.m7.2a"><mrow id="S5.p1.7.m7.2.2.2" xref="S5.p1.7.m7.2.2.3.cmml"><mo id="S5.p1.7.m7.2.2.2.3" xref="S5.p1.7.m7.2.2.3.cmml">(</mo><msub id="S5.p1.7.m7.1.1.1.1" xref="S5.p1.7.m7.1.1.1.1.cmml"><mi id="S5.p1.7.m7.1.1.1.1.2" xref="S5.p1.7.m7.1.1.1.1.2.cmml">v</mi><mn id="S5.p1.7.m7.1.1.1.1.3" xref="S5.p1.7.m7.1.1.1.1.3.cmml">0</mn></msub><mo id="S5.p1.7.m7.2.2.2.4" rspace="0.337em" xref="S5.p1.7.m7.2.2.3.cmml">,</mo><msub id="S5.p1.7.m7.2.2.2.2" xref="S5.p1.7.m7.2.2.2.2.cmml"><mi id="S5.p1.7.m7.2.2.2.2.2" xref="S5.p1.7.m7.2.2.2.2.2.cmml">p</mi><mrow id="S5.p1.7.m7.2.2.2.2.3" xref="S5.p1.7.m7.2.2.2.2.3.cmml"><mi id="S5.p1.7.m7.2.2.2.2.3.2" xref="S5.p1.7.m7.2.2.2.2.3.2.cmml">v</mi><mo id="S5.p1.7.m7.2.2.2.2.3.1" xref="S5.p1.7.m7.2.2.2.2.3.1.cmml">⁢</mo><mn id="S5.p1.7.m7.2.2.2.2.3.3" xref="S5.p1.7.m7.2.2.2.2.3.3.cmml">0</mn></mrow></msub><mo id="S5.p1.7.m7.2.2.2.5" xref="S5.p1.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.7.m7.2b"><interval closure="open" id="S5.p1.7.m7.2.2.3.cmml" xref="S5.p1.7.m7.2.2.2"><apply id="S5.p1.7.m7.1.1.1.1.cmml" xref="S5.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S5.p1.7.m7.1.1.1.1.1.cmml" xref="S5.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="S5.p1.7.m7.1.1.1.1.2.cmml" xref="S5.p1.7.m7.1.1.1.1.2">𝑣</ci><cn id="S5.p1.7.m7.1.1.1.1.3.cmml" type="integer" xref="S5.p1.7.m7.1.1.1.1.3">0</cn></apply><apply id="S5.p1.7.m7.2.2.2.2.cmml" xref="S5.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S5.p1.7.m7.2.2.2.2.1.cmml" xref="S5.p1.7.m7.2.2.2.2">subscript</csymbol><ci id="S5.p1.7.m7.2.2.2.2.2.cmml" xref="S5.p1.7.m7.2.2.2.2.2">𝑝</ci><apply id="S5.p1.7.m7.2.2.2.2.3.cmml" xref="S5.p1.7.m7.2.2.2.2.3"><times id="S5.p1.7.m7.2.2.2.2.3.1.cmml" xref="S5.p1.7.m7.2.2.2.2.3.1"></times><ci id="S5.p1.7.m7.2.2.2.2.3.2.cmml" xref="S5.p1.7.m7.2.2.2.2.3.2">𝑣</ci><cn id="S5.p1.7.m7.2.2.2.2.3.3.cmml" type="integer" xref="S5.p1.7.m7.2.2.2.2.3.3">0</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.7.m7.2c">\left(v_{0},\,p_{v0}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.7.m7.2d">( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_v 0 end_POSTSUBSCRIPT )</annotation></semantics></math> on the surface of section with <math alttext="u=0" class="ltx_Math" display="inline" id="S5.p1.8.m8.1"><semantics id="S5.p1.8.m8.1a"><mrow id="S5.p1.8.m8.1.1" xref="S5.p1.8.m8.1.1.cmml"><mi id="S5.p1.8.m8.1.1.2" xref="S5.p1.8.m8.1.1.2.cmml">u</mi><mo id="S5.p1.8.m8.1.1.1" xref="S5.p1.8.m8.1.1.1.cmml">=</mo><mn id="S5.p1.8.m8.1.1.3" xref="S5.p1.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.8.m8.1b"><apply id="S5.p1.8.m8.1.1.cmml" xref="S5.p1.8.m8.1.1"><eq id="S5.p1.8.m8.1.1.1.cmml" xref="S5.p1.8.m8.1.1.1"></eq><ci id="S5.p1.8.m8.1.1.2.cmml" xref="S5.p1.8.m8.1.1.2">𝑢</ci><cn id="S5.p1.8.m8.1.1.3.cmml" type="integer" xref="S5.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.8.m8.1c">u=0</annotation><annotation encoding="application/x-llamapun" id="S5.p1.8.m8.1d">italic_u = 0</annotation></semantics></math> and <math alttext="p_{u}=\sqrt{4-2\,V(0,v)-p_{v}^{2}}" class="ltx_Math" display="inline" id="S5.p1.9.m9.2"><semantics id="S5.p1.9.m9.2a"><mrow id="S5.p1.9.m9.2.3" xref="S5.p1.9.m9.2.3.cmml"><msub id="S5.p1.9.m9.2.3.2" xref="S5.p1.9.m9.2.3.2.cmml"><mi id="S5.p1.9.m9.2.3.2.2" xref="S5.p1.9.m9.2.3.2.2.cmml">p</mi><mi id="S5.p1.9.m9.2.3.2.3" xref="S5.p1.9.m9.2.3.2.3.cmml">u</mi></msub><mo id="S5.p1.9.m9.2.3.1" xref="S5.p1.9.m9.2.3.1.cmml">=</mo><msqrt id="S5.p1.9.m9.2.2" xref="S5.p1.9.m9.2.2.cmml"><mrow id="S5.p1.9.m9.2.2.2" xref="S5.p1.9.m9.2.2.2.cmml"><mn id="S5.p1.9.m9.2.2.2.4" xref="S5.p1.9.m9.2.2.2.4.cmml">4</mn><mo id="S5.p1.9.m9.2.2.2.3" xref="S5.p1.9.m9.2.2.2.3.cmml">−</mo><mrow id="S5.p1.9.m9.2.2.2.5" xref="S5.p1.9.m9.2.2.2.5.cmml"><mn id="S5.p1.9.m9.2.2.2.5.2" xref="S5.p1.9.m9.2.2.2.5.2.cmml">2</mn><mo id="S5.p1.9.m9.2.2.2.5.1" lspace="0.170em" xref="S5.p1.9.m9.2.2.2.5.1.cmml">⁢</mo><mi id="S5.p1.9.m9.2.2.2.5.3" xref="S5.p1.9.m9.2.2.2.5.3.cmml">V</mi><mo id="S5.p1.9.m9.2.2.2.5.1a" xref="S5.p1.9.m9.2.2.2.5.1.cmml">⁢</mo><mrow id="S5.p1.9.m9.2.2.2.5.4.2" xref="S5.p1.9.m9.2.2.2.5.4.1.cmml"><mo id="S5.p1.9.m9.2.2.2.5.4.2.1" stretchy="false" xref="S5.p1.9.m9.2.2.2.5.4.1.cmml">(</mo><mn id="S5.p1.9.m9.1.1.1.1" xref="S5.p1.9.m9.1.1.1.1.cmml">0</mn><mo id="S5.p1.9.m9.2.2.2.5.4.2.2" xref="S5.p1.9.m9.2.2.2.5.4.1.cmml">,</mo><mi id="S5.p1.9.m9.2.2.2.2" xref="S5.p1.9.m9.2.2.2.2.cmml">v</mi><mo id="S5.p1.9.m9.2.2.2.5.4.2.3" stretchy="false" xref="S5.p1.9.m9.2.2.2.5.4.1.cmml">)</mo></mrow></mrow><mo id="S5.p1.9.m9.2.2.2.3a" xref="S5.p1.9.m9.2.2.2.3.cmml">−</mo><msubsup id="S5.p1.9.m9.2.2.2.6" xref="S5.p1.9.m9.2.2.2.6.cmml"><mi id="S5.p1.9.m9.2.2.2.6.2.2" xref="S5.p1.9.m9.2.2.2.6.2.2.cmml">p</mi><mi id="S5.p1.9.m9.2.2.2.6.2.3" xref="S5.p1.9.m9.2.2.2.6.2.3.cmml">v</mi><mn id="S5.p1.9.m9.2.2.2.6.3" xref="S5.p1.9.m9.2.2.2.6.3.cmml">2</mn></msubsup></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.9.m9.2b"><apply id="S5.p1.9.m9.2.3.cmml" xref="S5.p1.9.m9.2.3"><eq id="S5.p1.9.m9.2.3.1.cmml" xref="S5.p1.9.m9.2.3.1"></eq><apply id="S5.p1.9.m9.2.3.2.cmml" xref="S5.p1.9.m9.2.3.2"><csymbol cd="ambiguous" id="S5.p1.9.m9.2.3.2.1.cmml" xref="S5.p1.9.m9.2.3.2">subscript</csymbol><ci id="S5.p1.9.m9.2.3.2.2.cmml" xref="S5.p1.9.m9.2.3.2.2">𝑝</ci><ci id="S5.p1.9.m9.2.3.2.3.cmml" xref="S5.p1.9.m9.2.3.2.3">𝑢</ci></apply><apply id="S5.p1.9.m9.2.2.cmml" xref="S5.p1.9.m9.2.2"><root id="S5.p1.9.m9.2.2a.cmml" xref="S5.p1.9.m9.2.2"></root><apply id="S5.p1.9.m9.2.2.2.cmml" xref="S5.p1.9.m9.2.2.2"><minus id="S5.p1.9.m9.2.2.2.3.cmml" xref="S5.p1.9.m9.2.2.2.3"></minus><cn id="S5.p1.9.m9.2.2.2.4.cmml" type="integer" xref="S5.p1.9.m9.2.2.2.4">4</cn><apply id="S5.p1.9.m9.2.2.2.5.cmml" xref="S5.p1.9.m9.2.2.2.5"><times id="S5.p1.9.m9.2.2.2.5.1.cmml" xref="S5.p1.9.m9.2.2.2.5.1"></times><cn id="S5.p1.9.m9.2.2.2.5.2.cmml" type="integer" xref="S5.p1.9.m9.2.2.2.5.2">2</cn><ci id="S5.p1.9.m9.2.2.2.5.3.cmml" xref="S5.p1.9.m9.2.2.2.5.3">𝑉</ci><interval closure="open" id="S5.p1.9.m9.2.2.2.5.4.1.cmml" xref="S5.p1.9.m9.2.2.2.5.4.2"><cn id="S5.p1.9.m9.1.1.1.1.cmml" type="integer" xref="S5.p1.9.m9.1.1.1.1">0</cn><ci id="S5.p1.9.m9.2.2.2.2.cmml" xref="S5.p1.9.m9.2.2.2.2">𝑣</ci></interval></apply><apply id="S5.p1.9.m9.2.2.2.6.cmml" xref="S5.p1.9.m9.2.2.2.6"><csymbol cd="ambiguous" id="S5.p1.9.m9.2.2.2.6.1.cmml" xref="S5.p1.9.m9.2.2.2.6">superscript</csymbol><apply id="S5.p1.9.m9.2.2.2.6.2.cmml" xref="S5.p1.9.m9.2.2.2.6"><csymbol cd="ambiguous" id="S5.p1.9.m9.2.2.2.6.2.1.cmml" xref="S5.p1.9.m9.2.2.2.6">subscript</csymbol><ci id="S5.p1.9.m9.2.2.2.6.2.2.cmml" xref="S5.p1.9.m9.2.2.2.6.2.2">𝑝</ci><ci id="S5.p1.9.m9.2.2.2.6.2.3.cmml" xref="S5.p1.9.m9.2.2.2.6.2.3">𝑣</ci></apply><cn id="S5.p1.9.m9.2.2.2.6.3.cmml" type="integer" xref="S5.p1.9.m9.2.2.2.6.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.9.m9.2c">p_{u}=\sqrt{4-2\,V(0,v)-p_{v}^{2}}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.9.m9.2d">italic_p start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT = square-root start_ARG 4 - 2 italic_V ( 0 , italic_v ) - italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>, where we have chosen <math alttext="p_{u}>0" class="ltx_Math" display="inline" id="S5.p1.10.m10.1"><semantics id="S5.p1.10.m10.1a"><mrow id="S5.p1.10.m10.1.1" xref="S5.p1.10.m10.1.1.cmml"><msub id="S5.p1.10.m10.1.1.2" xref="S5.p1.10.m10.1.1.2.cmml"><mi id="S5.p1.10.m10.1.1.2.2" xref="S5.p1.10.m10.1.1.2.2.cmml">p</mi><mi id="S5.p1.10.m10.1.1.2.3" xref="S5.p1.10.m10.1.1.2.3.cmml">u</mi></msub><mo id="S5.p1.10.m10.1.1.1" xref="S5.p1.10.m10.1.1.1.cmml">></mo><mn id="S5.p1.10.m10.1.1.3" xref="S5.p1.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.10.m10.1b"><apply id="S5.p1.10.m10.1.1.cmml" xref="S5.p1.10.m10.1.1"><gt id="S5.p1.10.m10.1.1.1.cmml" xref="S5.p1.10.m10.1.1.1"></gt><apply id="S5.p1.10.m10.1.1.2.cmml" xref="S5.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.p1.10.m10.1.1.2.1.cmml" xref="S5.p1.10.m10.1.1.2">subscript</csymbol><ci id="S5.p1.10.m10.1.1.2.2.cmml" xref="S5.p1.10.m10.1.1.2.2">𝑝</ci><ci id="S5.p1.10.m10.1.1.2.3.cmml" xref="S5.p1.10.m10.1.1.2.3">𝑢</ci></apply><cn id="S5.p1.10.m10.1.1.3.cmml" type="integer" xref="S5.p1.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.10.m10.1c">p_{u}>0</annotation><annotation encoding="application/x-llamapun" id="S5.p1.10.m10.1d">italic_p start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT > 0</annotation></semantics></math>. The trajectory is thus launched into the region <math alttext="u>0" class="ltx_Math" display="inline" id="S5.p1.11.m11.1"><semantics id="S5.p1.11.m11.1a"><mrow id="S5.p1.11.m11.1.1" xref="S5.p1.11.m11.1.1.cmml"><mi id="S5.p1.11.m11.1.1.2" xref="S5.p1.11.m11.1.1.2.cmml">u</mi><mo id="S5.p1.11.m11.1.1.1" xref="S5.p1.11.m11.1.1.1.cmml">></mo><mn id="S5.p1.11.m11.1.1.3" xref="S5.p1.11.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.11.m11.1b"><apply id="S5.p1.11.m11.1.1.cmml" xref="S5.p1.11.m11.1.1"><gt id="S5.p1.11.m11.1.1.1.cmml" xref="S5.p1.11.m11.1.1.1"></gt><ci id="S5.p1.11.m11.1.1.2.cmml" xref="S5.p1.11.m11.1.1.2">𝑢</ci><cn id="S5.p1.11.m11.1.1.3.cmml" type="integer" xref="S5.p1.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.11.m11.1c">u>0</annotation><annotation encoding="application/x-llamapun" id="S5.p1.11.m11.1d">italic_u > 0</annotation></semantics></math>. When the trajectory intersects the surface of section again, the map returns the coordinates <math alttext="\left(v_{1},\,p_{v1}\right)" class="ltx_Math" display="inline" id="S5.p1.12.m12.2"><semantics id="S5.p1.12.m12.2a"><mrow id="S5.p1.12.m12.2.2.2" xref="S5.p1.12.m12.2.2.3.cmml"><mo id="S5.p1.12.m12.2.2.2.3" xref="S5.p1.12.m12.2.2.3.cmml">(</mo><msub id="S5.p1.12.m12.1.1.1.1" xref="S5.p1.12.m12.1.1.1.1.cmml"><mi id="S5.p1.12.m12.1.1.1.1.2" xref="S5.p1.12.m12.1.1.1.1.2.cmml">v</mi><mn id="S5.p1.12.m12.1.1.1.1.3" xref="S5.p1.12.m12.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.p1.12.m12.2.2.2.4" rspace="0.337em" xref="S5.p1.12.m12.2.2.3.cmml">,</mo><msub id="S5.p1.12.m12.2.2.2.2" xref="S5.p1.12.m12.2.2.2.2.cmml"><mi id="S5.p1.12.m12.2.2.2.2.2" xref="S5.p1.12.m12.2.2.2.2.2.cmml">p</mi><mrow id="S5.p1.12.m12.2.2.2.2.3" xref="S5.p1.12.m12.2.2.2.2.3.cmml"><mi id="S5.p1.12.m12.2.2.2.2.3.2" xref="S5.p1.12.m12.2.2.2.2.3.2.cmml">v</mi><mo id="S5.p1.12.m12.2.2.2.2.3.1" xref="S5.p1.12.m12.2.2.2.2.3.1.cmml">⁢</mo><mn id="S5.p1.12.m12.2.2.2.2.3.3" xref="S5.p1.12.m12.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.p1.12.m12.2.2.2.5" xref="S5.p1.12.m12.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.12.m12.2b"><interval closure="open" id="S5.p1.12.m12.2.2.3.cmml" xref="S5.p1.12.m12.2.2.2"><apply id="S5.p1.12.m12.1.1.1.1.cmml" xref="S5.p1.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S5.p1.12.m12.1.1.1.1.1.cmml" xref="S5.p1.12.m12.1.1.1.1">subscript</csymbol><ci id="S5.p1.12.m12.1.1.1.1.2.cmml" xref="S5.p1.12.m12.1.1.1.1.2">𝑣</ci><cn id="S5.p1.12.m12.1.1.1.1.3.cmml" type="integer" xref="S5.p1.12.m12.1.1.1.1.3">1</cn></apply><apply id="S5.p1.12.m12.2.2.2.2.cmml" xref="S5.p1.12.m12.2.2.2.2"><csymbol cd="ambiguous" id="S5.p1.12.m12.2.2.2.2.1.cmml" xref="S5.p1.12.m12.2.2.2.2">subscript</csymbol><ci id="S5.p1.12.m12.2.2.2.2.2.cmml" xref="S5.p1.12.m12.2.2.2.2.2">𝑝</ci><apply id="S5.p1.12.m12.2.2.2.2.3.cmml" xref="S5.p1.12.m12.2.2.2.2.3"><times id="S5.p1.12.m12.2.2.2.2.3.1.cmml" xref="S5.p1.12.m12.2.2.2.2.3.1"></times><ci id="S5.p1.12.m12.2.2.2.2.3.2.cmml" xref="S5.p1.12.m12.2.2.2.2.3.2">𝑣</ci><cn id="S5.p1.12.m12.2.2.2.2.3.3.cmml" type="integer" xref="S5.p1.12.m12.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.12.m12.2c">\left(v_{1},\,p_{v1}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.12.m12.2d">( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_v 1 end_POSTSUBSCRIPT )</annotation></semantics></math> of the intersection point. Note that due to symmetry, the map does not depend on our initial choice <math alttext="p_{u}>0" class="ltx_Math" display="inline" id="S5.p1.13.m13.1"><semantics id="S5.p1.13.m13.1a"><mrow id="S5.p1.13.m13.1.1" xref="S5.p1.13.m13.1.1.cmml"><msub id="S5.p1.13.m13.1.1.2" xref="S5.p1.13.m13.1.1.2.cmml"><mi id="S5.p1.13.m13.1.1.2.2" xref="S5.p1.13.m13.1.1.2.2.cmml">p</mi><mi id="S5.p1.13.m13.1.1.2.3" xref="S5.p1.13.m13.1.1.2.3.cmml">u</mi></msub><mo id="S5.p1.13.m13.1.1.1" xref="S5.p1.13.m13.1.1.1.cmml">></mo><mn id="S5.p1.13.m13.1.1.3" xref="S5.p1.13.m13.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.13.m13.1b"><apply id="S5.p1.13.m13.1.1.cmml" xref="S5.p1.13.m13.1.1"><gt id="S5.p1.13.m13.1.1.1.cmml" xref="S5.p1.13.m13.1.1.1"></gt><apply id="S5.p1.13.m13.1.1.2.cmml" xref="S5.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S5.p1.13.m13.1.1.2.1.cmml" xref="S5.p1.13.m13.1.1.2">subscript</csymbol><ci id="S5.p1.13.m13.1.1.2.2.cmml" xref="S5.p1.13.m13.1.1.2.2">𝑝</ci><ci id="S5.p1.13.m13.1.1.2.3.cmml" xref="S5.p1.13.m13.1.1.2.3">𝑢</ci></apply><cn id="S5.p1.13.m13.1.1.3.cmml" type="integer" xref="S5.p1.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.13.m13.1c">p_{u}>0</annotation><annotation encoding="application/x-llamapun" id="S5.p1.13.m13.1d">italic_p start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT > 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">All periodic orbits of the continuous-time system in the full phase space pass through the surface of section. Thus, it is sufficient to find the periodic orbits of the discrete map <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">caligraphic_M</annotation></semantics></math>. Once these discrete orbits are acquired, one can recover the continuous-time orbits by integrating the discrete points through the full phase space.</p> </div> <figure class="ltx_figure" id="S5.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="476" id="S5.F2.g1" src="extracted/6294607/resonanceZoneDiscreteMonteLabeled.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S5.F2.4.2.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S5.F2.2.1" style="font-size:90%;">Initial ensemble for discrete-time Monte Carlo simulations. Initial ensemble shown in blue with <math alttext="10^{7}" class="ltx_Math" display="inline" id="S5.F2.2.1.m1.1"><semantics id="S5.F2.2.1.m1.1b"><msup id="S5.F2.2.1.m1.1.1" xref="S5.F2.2.1.m1.1.1.cmml"><mn id="S5.F2.2.1.m1.1.1.2" xref="S5.F2.2.1.m1.1.1.2.cmml">10</mn><mn id="S5.F2.2.1.m1.1.1.3" xref="S5.F2.2.1.m1.1.1.3.cmml">7</mn></msup><annotation-xml encoding="MathML-Content" id="S5.F2.2.1.m1.1c"><apply id="S5.F2.2.1.m1.1.1.cmml" xref="S5.F2.2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.F2.2.1.m1.1.1.1.cmml" xref="S5.F2.2.1.m1.1.1">superscript</csymbol><cn id="S5.F2.2.1.m1.1.1.2.cmml" type="integer" xref="S5.F2.2.1.m1.1.1.2">10</cn><cn id="S5.F2.2.1.m1.1.1.3.cmml" type="integer" xref="S5.F2.2.1.m1.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F2.2.1.m1.1d">10^{7}</annotation><annotation encoding="application/x-llamapun" id="S5.F2.2.1.m1.1e">10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT</annotation></semantics></math> initial points. Resonance zone shown in gray bounded by stable and unstable manifolds. (See also Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a.) Escape boundary drawn as a black square around the resonance zone.</span></figcaption> </figure> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.9">To study the dynamics of the discrete map, we conduct discrete-time Monte Carlo simulations using <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><ci id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">caligraphic_M</annotation></semantics></math> to compute the discrete escape rate <math alttext="\gamma_{d}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><msub id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mi id="S5.p3.2.m2.1.1.2" xref="S5.p3.2.m2.1.1.2.cmml">γ</mi><mi id="S5.p3.2.m2.1.1.3" xref="S5.p3.2.m2.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.1.cmml" xref="S5.p3.2.m2.1.1">subscript</csymbol><ci id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2">𝛾</ci><ci id="S5.p3.2.m2.1.1.3.cmml" xref="S5.p3.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\gamma_{d}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">italic_γ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> at given values of <math alttext="B" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mi id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><ci id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">italic_B</annotation></semantics></math> and <math alttext="E" class="ltx_Math" display="inline" id="S5.p3.4.m4.1"><semantics id="S5.p3.4.m4.1a"><mi id="S5.p3.4.m4.1.1" xref="S5.p3.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S5.p3.4.m4.1b"><ci id="S5.p3.4.m4.1.1.cmml" xref="S5.p3.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S5.p3.4.m4.1d">italic_E</annotation></semantics></math>. We use an initial ensemble of <math alttext="10^{7}" class="ltx_Math" display="inline" id="S5.p3.5.m5.1"><semantics id="S5.p3.5.m5.1a"><msup id="S5.p3.5.m5.1.1" xref="S5.p3.5.m5.1.1.cmml"><mn id="S5.p3.5.m5.1.1.2" xref="S5.p3.5.m5.1.1.2.cmml">10</mn><mn id="S5.p3.5.m5.1.1.3" xref="S5.p3.5.m5.1.1.3.cmml">7</mn></msup><annotation-xml encoding="MathML-Content" id="S5.p3.5.m5.1b"><apply id="S5.p3.5.m5.1.1.cmml" xref="S5.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S5.p3.5.m5.1.1.1.cmml" xref="S5.p3.5.m5.1.1">superscript</csymbol><cn id="S5.p3.5.m5.1.1.2.cmml" type="integer" xref="S5.p3.5.m5.1.1.2">10</cn><cn id="S5.p3.5.m5.1.1.3.cmml" type="integer" xref="S5.p3.5.m5.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.5.m5.1c">10^{7}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.5.m5.1d">10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT</annotation></semantics></math> points arranged in a disk in the <math alttext="vp_{v}" class="ltx_Math" display="inline" id="S5.p3.6.m6.1"><semantics id="S5.p3.6.m6.1a"><mrow id="S5.p3.6.m6.1.1" xref="S5.p3.6.m6.1.1.cmml"><mi id="S5.p3.6.m6.1.1.2" xref="S5.p3.6.m6.1.1.2.cmml">v</mi><mo id="S5.p3.6.m6.1.1.1" xref="S5.p3.6.m6.1.1.1.cmml">⁢</mo><msub id="S5.p3.6.m6.1.1.3" xref="S5.p3.6.m6.1.1.3.cmml"><mi id="S5.p3.6.m6.1.1.3.2" xref="S5.p3.6.m6.1.1.3.2.cmml">p</mi><mi id="S5.p3.6.m6.1.1.3.3" xref="S5.p3.6.m6.1.1.3.3.cmml">v</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.6.m6.1b"><apply id="S5.p3.6.m6.1.1.cmml" xref="S5.p3.6.m6.1.1"><times id="S5.p3.6.m6.1.1.1.cmml" xref="S5.p3.6.m6.1.1.1"></times><ci id="S5.p3.6.m6.1.1.2.cmml" xref="S5.p3.6.m6.1.1.2">𝑣</ci><apply id="S5.p3.6.m6.1.1.3.cmml" xref="S5.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.p3.6.m6.1.1.3.1.cmml" xref="S5.p3.6.m6.1.1.3">subscript</csymbol><ci id="S5.p3.6.m6.1.1.3.2.cmml" xref="S5.p3.6.m6.1.1.3.2">𝑝</ci><ci id="S5.p3.6.m6.1.1.3.3.cmml" xref="S5.p3.6.m6.1.1.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.6.m6.1c">vp_{v}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.6.m6.1d">italic_v italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> plane, centered at the origin and contained completely within the resonance zone shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S5.F2" title="Figure 2 ‣ V Surface of Section and Discrete-Time Monte Carlo ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">2</span></a>. When points are mapped outside the resonance zone, they will quickly escape to infinity. A box is drawn around the resonance zone, and escape is recorded when a point is mapped outside this box. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S3.F1" title="Figure 1 ‣ III Escape Rate from Classical Trajectory Monte Carlo ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">1</span></a>b plots the number of surviving points as a function of iterate computed at <math alttext="E=1.0" class="ltx_Math" display="inline" id="S5.p3.7.m7.1"><semantics id="S5.p3.7.m7.1a"><mrow id="S5.p3.7.m7.1.1" xref="S5.p3.7.m7.1.1.cmml"><mi id="S5.p3.7.m7.1.1.2" xref="S5.p3.7.m7.1.1.2.cmml">E</mi><mo id="S5.p3.7.m7.1.1.1" xref="S5.p3.7.m7.1.1.1.cmml">=</mo><mn id="S5.p3.7.m7.1.1.3" xref="S5.p3.7.m7.1.1.3.cmml">1.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.7.m7.1b"><apply id="S5.p3.7.m7.1.1.cmml" xref="S5.p3.7.m7.1.1"><eq id="S5.p3.7.m7.1.1.1.cmml" xref="S5.p3.7.m7.1.1.1"></eq><ci id="S5.p3.7.m7.1.1.2.cmml" xref="S5.p3.7.m7.1.1.2">𝐸</ci><cn id="S5.p3.7.m7.1.1.3.cmml" type="float" xref="S5.p3.7.m7.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.7.m7.1c">E=1.0</annotation><annotation encoding="application/x-llamapun" id="S5.p3.7.m7.1d">italic_E = 1.0</annotation></semantics></math> and <math alttext="B=3.5" class="ltx_Math" display="inline" id="S5.p3.8.m8.1"><semantics id="S5.p3.8.m8.1a"><mrow id="S5.p3.8.m8.1.1" xref="S5.p3.8.m8.1.1.cmml"><mi id="S5.p3.8.m8.1.1.2" xref="S5.p3.8.m8.1.1.2.cmml">B</mi><mo id="S5.p3.8.m8.1.1.1" xref="S5.p3.8.m8.1.1.1.cmml">=</mo><mn id="S5.p3.8.m8.1.1.3" xref="S5.p3.8.m8.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.8.m8.1b"><apply id="S5.p3.8.m8.1.1.cmml" xref="S5.p3.8.m8.1.1"><eq id="S5.p3.8.m8.1.1.1.cmml" xref="S5.p3.8.m8.1.1.1"></eq><ci id="S5.p3.8.m8.1.1.2.cmml" xref="S5.p3.8.m8.1.1.2">𝐵</ci><cn id="S5.p3.8.m8.1.1.3.cmml" type="float" xref="S5.p3.8.m8.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.8.m8.1c">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S5.p3.8.m8.1d">italic_B = 3.5</annotation></semantics></math>. It shows clear exponential decay, with decay rate <math alttext="\gamma_{d}=0.8456\pm 0.012" class="ltx_Math" display="inline" id="S5.p3.9.m9.1"><semantics id="S5.p3.9.m9.1a"><mrow id="S5.p3.9.m9.1.1" xref="S5.p3.9.m9.1.1.cmml"><msub id="S5.p3.9.m9.1.1.2" xref="S5.p3.9.m9.1.1.2.cmml"><mi id="S5.p3.9.m9.1.1.2.2" xref="S5.p3.9.m9.1.1.2.2.cmml">γ</mi><mi id="S5.p3.9.m9.1.1.2.3" xref="S5.p3.9.m9.1.1.2.3.cmml">d</mi></msub><mo id="S5.p3.9.m9.1.1.1" xref="S5.p3.9.m9.1.1.1.cmml">=</mo><mrow id="S5.p3.9.m9.1.1.3" xref="S5.p3.9.m9.1.1.3.cmml"><mn id="S5.p3.9.m9.1.1.3.2" xref="S5.p3.9.m9.1.1.3.2.cmml">0.8456</mn><mo id="S5.p3.9.m9.1.1.3.1" xref="S5.p3.9.m9.1.1.3.1.cmml">±</mo><mn id="S5.p3.9.m9.1.1.3.3" xref="S5.p3.9.m9.1.1.3.3.cmml">0.012</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.9.m9.1b"><apply id="S5.p3.9.m9.1.1.cmml" xref="S5.p3.9.m9.1.1"><eq id="S5.p3.9.m9.1.1.1.cmml" xref="S5.p3.9.m9.1.1.1"></eq><apply id="S5.p3.9.m9.1.1.2.cmml" xref="S5.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S5.p3.9.m9.1.1.2.1.cmml" xref="S5.p3.9.m9.1.1.2">subscript</csymbol><ci id="S5.p3.9.m9.1.1.2.2.cmml" xref="S5.p3.9.m9.1.1.2.2">𝛾</ci><ci id="S5.p3.9.m9.1.1.2.3.cmml" xref="S5.p3.9.m9.1.1.2.3">𝑑</ci></apply><apply id="S5.p3.9.m9.1.1.3.cmml" xref="S5.p3.9.m9.1.1.3"><csymbol cd="latexml" id="S5.p3.9.m9.1.1.3.1.cmml" xref="S5.p3.9.m9.1.1.3.1">plus-or-minus</csymbol><cn id="S5.p3.9.m9.1.1.3.2.cmml" type="float" xref="S5.p3.9.m9.1.1.3.2">0.8456</cn><cn id="S5.p3.9.m9.1.1.3.3.cmml" type="float" xref="S5.p3.9.m9.1.1.3.3">0.012</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.9.m9.1c">\gamma_{d}=0.8456\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S5.p3.9.m9.1d">italic_γ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0.8456 ± 0.012</annotation></semantics></math> in units of inverse iterate. We use the same method as the continuous-time case to generate the fit and extract the decay rate. This discrete decay rate can also be computed using the discrete periodic orbit method above, which we will demonstrate below in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7" title="VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">VII</span></a>.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span>Computing Periodic Orbits via Phase Space Partitioning</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.7">In order to compute periodic orbits, we must generate accurate initial guesses to use in a Newton’s method solver. To generate this set of guesses, we will partition the <math alttext="vp_{v}" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mrow id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mi id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2.cmml">v</mi><mo id="S6.p1.1.m1.1.1.1" xref="S6.p1.1.m1.1.1.1.cmml">⁢</mo><msub id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml"><mi id="S6.p1.1.m1.1.1.3.2" xref="S6.p1.1.m1.1.1.3.2.cmml">p</mi><mi id="S6.p1.1.m1.1.1.3.3" xref="S6.p1.1.m1.1.1.3.3.cmml">v</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><times id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1.1"></times><ci id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">𝑣</ci><apply id="S6.p1.1.m1.1.1.3.cmml" xref="S6.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.3.1.cmml" xref="S6.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.p1.1.m1.1.1.3.2.cmml" xref="S6.p1.1.m1.1.1.3.2">𝑝</ci><ci id="S6.p1.1.m1.1.1.3.3.cmml" xref="S6.p1.1.m1.1.1.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">vp_{v}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">italic_v italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> phase space to produce a discrete Markov process between the partition domains. We can do this because at <math alttext="E=1" class="ltx_Math" display="inline" id="S6.p1.2.m2.1"><semantics id="S6.p1.2.m2.1a"><mrow id="S6.p1.2.m2.1.1" xref="S6.p1.2.m2.1.1.cmml"><mi id="S6.p1.2.m2.1.1.2" xref="S6.p1.2.m2.1.1.2.cmml">E</mi><mo id="S6.p1.2.m2.1.1.1" xref="S6.p1.2.m2.1.1.1.cmml">=</mo><mn id="S6.p1.2.m2.1.1.3" xref="S6.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.1b"><apply id="S6.p1.2.m2.1.1.cmml" xref="S6.p1.2.m2.1.1"><eq id="S6.p1.2.m2.1.1.1.cmml" xref="S6.p1.2.m2.1.1.1"></eq><ci id="S6.p1.2.m2.1.1.2.cmml" xref="S6.p1.2.m2.1.1.2">𝐸</ci><cn id="S6.p1.2.m2.1.1.3.cmml" type="integer" xref="S6.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.1c">E=1</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.1d">italic_E = 1</annotation></semantics></math>, <math alttext="B=3.5" class="ltx_Math" display="inline" id="S6.p1.3.m3.1"><semantics id="S6.p1.3.m3.1a"><mrow id="S6.p1.3.m3.1.1" xref="S6.p1.3.m3.1.1.cmml"><mi id="S6.p1.3.m3.1.1.2" xref="S6.p1.3.m3.1.1.2.cmml">B</mi><mo id="S6.p1.3.m3.1.1.1" xref="S6.p1.3.m3.1.1.1.cmml">=</mo><mn id="S6.p1.3.m3.1.1.3" xref="S6.p1.3.m3.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.3.m3.1b"><apply id="S6.p1.3.m3.1.1.cmml" xref="S6.p1.3.m3.1.1"><eq id="S6.p1.3.m3.1.1.1.cmml" xref="S6.p1.3.m3.1.1.1"></eq><ci id="S6.p1.3.m3.1.1.2.cmml" xref="S6.p1.3.m3.1.1.2">𝐵</ci><cn id="S6.p1.3.m3.1.1.3.cmml" type="float" xref="S6.p1.3.m3.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.3.m3.1c">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S6.p1.3.m3.1d">italic_B = 3.5</annotation></semantics></math> the dynamics on the surface of section are entirely hyperbolic with no invariant tori. There are two hyperbolic fixed points <math alttext="z_{l}" class="ltx_Math" display="inline" id="S6.p1.4.m4.1"><semantics id="S6.p1.4.m4.1a"><msub id="S6.p1.4.m4.1.1" xref="S6.p1.4.m4.1.1.cmml"><mi id="S6.p1.4.m4.1.1.2" xref="S6.p1.4.m4.1.1.2.cmml">z</mi><mi id="S6.p1.4.m4.1.1.3" xref="S6.p1.4.m4.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p1.4.m4.1b"><apply id="S6.p1.4.m4.1.1.cmml" xref="S6.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.p1.4.m4.1.1.1.cmml" xref="S6.p1.4.m4.1.1">subscript</csymbol><ci id="S6.p1.4.m4.1.1.2.cmml" xref="S6.p1.4.m4.1.1.2">𝑧</ci><ci id="S6.p1.4.m4.1.1.3.cmml" xref="S6.p1.4.m4.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.4.m4.1c">z_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.4.m4.1d">italic_z start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="z_{r}" class="ltx_Math" display="inline" id="S6.p1.5.m5.1"><semantics id="S6.p1.5.m5.1a"><msub id="S6.p1.5.m5.1.1" xref="S6.p1.5.m5.1.1.cmml"><mi id="S6.p1.5.m5.1.1.2" xref="S6.p1.5.m5.1.1.2.cmml">z</mi><mi id="S6.p1.5.m5.1.1.3" xref="S6.p1.5.m5.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p1.5.m5.1b"><apply id="S6.p1.5.m5.1.1.cmml" xref="S6.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S6.p1.5.m5.1.1.1.cmml" xref="S6.p1.5.m5.1.1">subscript</csymbol><ci id="S6.p1.5.m5.1.1.2.cmml" xref="S6.p1.5.m5.1.1.2">𝑧</ci><ci id="S6.p1.5.m5.1.1.3.cmml" xref="S6.p1.5.m5.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.5.m5.1c">z_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.5.m5.1d">italic_z start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> on the <math alttext="v" class="ltx_Math" display="inline" id="S6.p1.6.m6.1"><semantics id="S6.p1.6.m6.1a"><mi id="S6.p1.6.m6.1.1" xref="S6.p1.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S6.p1.6.m6.1b"><ci id="S6.p1.6.m6.1.1.cmml" xref="S6.p1.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S6.p1.6.m6.1d">italic_v</annotation></semantics></math> axis of the surface of section, located symmetrically about the <math alttext="p_{v}" class="ltx_Math" display="inline" id="S6.p1.7.m7.1"><semantics id="S6.p1.7.m7.1a"><msub id="S6.p1.7.m7.1.1" xref="S6.p1.7.m7.1.1.cmml"><mi id="S6.p1.7.m7.1.1.2" xref="S6.p1.7.m7.1.1.2.cmml">p</mi><mi id="S6.p1.7.m7.1.1.3" xref="S6.p1.7.m7.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p1.7.m7.1b"><apply id="S6.p1.7.m7.1.1.cmml" xref="S6.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S6.p1.7.m7.1.1.1.cmml" xref="S6.p1.7.m7.1.1">subscript</csymbol><ci id="S6.p1.7.m7.1.1.2.cmml" xref="S6.p1.7.m7.1.1.2">𝑝</ci><ci id="S6.p1.7.m7.1.1.3.cmml" xref="S6.p1.7.m7.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.7.m7.1c">p_{v}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.7.m7.1d">italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> axis, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a. Emanating from those fixed points are one-dimensional stable (red) and unstable (blue) manifolds. All trajectories on a stable manifold are mapped towards the fixed point it emanates from. Conversely, all trajectories on an unstable manifold are mapped away from the fixed point. The stable and unstable manifolds are infinitely long, forming a complicated structure called a heteroclinic tangle, which describes the transport of points in phase space. We will construct the phase space partition from segments of these stable and unstable manifolds.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.6">Points where the stable and unstable manifolds intersect such that the open intervals of the manifolds up to that point do not intersect elsewhere are referred to as <span class="ltx_text ltx_font_italic" id="S6.p2.6.1">primary intersection points</span>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a shows two primary intersection points <math alttext="p_{0}" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><msub id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml"><mi id="S6.p2.1.m1.1.1.2" xref="S6.p2.1.m1.1.1.2.cmml">p</mi><mn id="S6.p2.1.m1.1.1.3" xref="S6.p2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><apply id="S6.p2.1.m1.1.1.cmml" xref="S6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.p2.1.m1.1.1.1.cmml" xref="S6.p2.1.m1.1.1">subscript</csymbol><ci id="S6.p2.1.m1.1.1.2.cmml" xref="S6.p2.1.m1.1.1.2">𝑝</ci><cn id="S6.p2.1.m1.1.1.3.cmml" type="integer" xref="S6.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">p_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tilde{p}_{0}" class="ltx_Math" display="inline" id="S6.p2.2.m2.1"><semantics id="S6.p2.2.m2.1a"><msub id="S6.p2.2.m2.1.1" xref="S6.p2.2.m2.1.1.cmml"><mover accent="true" id="S6.p2.2.m2.1.1.2" xref="S6.p2.2.m2.1.1.2.cmml"><mi id="S6.p2.2.m2.1.1.2.2" xref="S6.p2.2.m2.1.1.2.2.cmml">p</mi><mo id="S6.p2.2.m2.1.1.2.1" xref="S6.p2.2.m2.1.1.2.1.cmml">~</mo></mover><mn id="S6.p2.2.m2.1.1.3" xref="S6.p2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.2.m2.1b"><apply id="S6.p2.2.m2.1.1.cmml" xref="S6.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.p2.2.m2.1.1.1.cmml" xref="S6.p2.2.m2.1.1">subscript</csymbol><apply id="S6.p2.2.m2.1.1.2.cmml" xref="S6.p2.2.m2.1.1.2"><ci id="S6.p2.2.m2.1.1.2.1.cmml" xref="S6.p2.2.m2.1.1.2.1">~</ci><ci id="S6.p2.2.m2.1.1.2.2.cmml" xref="S6.p2.2.m2.1.1.2.2">𝑝</ci></apply><cn id="S6.p2.2.m2.1.1.3.cmml" type="integer" xref="S6.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.2.m2.1c">\tilde{p}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.2.m2.1d">over~ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and their forward iterates <math alttext="p_{1}" class="ltx_Math" display="inline" id="S6.p2.3.m3.1"><semantics id="S6.p2.3.m3.1a"><msub id="S6.p2.3.m3.1.1" xref="S6.p2.3.m3.1.1.cmml"><mi id="S6.p2.3.m3.1.1.2" xref="S6.p2.3.m3.1.1.2.cmml">p</mi><mn id="S6.p2.3.m3.1.1.3" xref="S6.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.3.m3.1b"><apply id="S6.p2.3.m3.1.1.cmml" xref="S6.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S6.p2.3.m3.1.1.1.cmml" xref="S6.p2.3.m3.1.1">subscript</csymbol><ci id="S6.p2.3.m3.1.1.2.cmml" xref="S6.p2.3.m3.1.1.2">𝑝</ci><cn id="S6.p2.3.m3.1.1.3.cmml" type="integer" xref="S6.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.3.m3.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.3.m3.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tilde{p}_{1}" class="ltx_Math" display="inline" id="S6.p2.4.m4.1"><semantics id="S6.p2.4.m4.1a"><msub id="S6.p2.4.m4.1.1" xref="S6.p2.4.m4.1.1.cmml"><mover accent="true" id="S6.p2.4.m4.1.1.2" xref="S6.p2.4.m4.1.1.2.cmml"><mi id="S6.p2.4.m4.1.1.2.2" xref="S6.p2.4.m4.1.1.2.2.cmml">p</mi><mo id="S6.p2.4.m4.1.1.2.1" xref="S6.p2.4.m4.1.1.2.1.cmml">~</mo></mover><mn id="S6.p2.4.m4.1.1.3" xref="S6.p2.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.4.m4.1b"><apply id="S6.p2.4.m4.1.1.cmml" xref="S6.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S6.p2.4.m4.1.1.1.cmml" xref="S6.p2.4.m4.1.1">subscript</csymbol><apply id="S6.p2.4.m4.1.1.2.cmml" xref="S6.p2.4.m4.1.1.2"><ci id="S6.p2.4.m4.1.1.2.1.cmml" xref="S6.p2.4.m4.1.1.2.1">~</ci><ci id="S6.p2.4.m4.1.1.2.2.cmml" xref="S6.p2.4.m4.1.1.2.2">𝑝</ci></apply><cn id="S6.p2.4.m4.1.1.3.cmml" type="integer" xref="S6.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.4.m4.1c">\tilde{p}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.4.m4.1d">over~ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. The intervals of the two stable and unstable manifolds up to the primary intersection points <math alttext="p_{0}" class="ltx_Math" display="inline" id="S6.p2.5.m5.1"><semantics id="S6.p2.5.m5.1a"><msub id="S6.p2.5.m5.1.1" xref="S6.p2.5.m5.1.1.cmml"><mi id="S6.p2.5.m5.1.1.2" xref="S6.p2.5.m5.1.1.2.cmml">p</mi><mn id="S6.p2.5.m5.1.1.3" xref="S6.p2.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.5.m5.1b"><apply id="S6.p2.5.m5.1.1.cmml" xref="S6.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S6.p2.5.m5.1.1.1.cmml" xref="S6.p2.5.m5.1.1">subscript</csymbol><ci id="S6.p2.5.m5.1.1.2.cmml" xref="S6.p2.5.m5.1.1.2">𝑝</ci><cn id="S6.p2.5.m5.1.1.3.cmml" type="integer" xref="S6.p2.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.5.m5.1c">p_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.5.m5.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tilde{p}_{0}" class="ltx_Math" display="inline" id="S6.p2.6.m6.1"><semantics id="S6.p2.6.m6.1a"><msub id="S6.p2.6.m6.1.1" xref="S6.p2.6.m6.1.1.cmml"><mover accent="true" id="S6.p2.6.m6.1.1.2" xref="S6.p2.6.m6.1.1.2.cmml"><mi id="S6.p2.6.m6.1.1.2.2" xref="S6.p2.6.m6.1.1.2.2.cmml">p</mi><mo id="S6.p2.6.m6.1.1.2.1" xref="S6.p2.6.m6.1.1.2.1.cmml">~</mo></mover><mn id="S6.p2.6.m6.1.1.3" xref="S6.p2.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.6.m6.1b"><apply id="S6.p2.6.m6.1.1.cmml" xref="S6.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.p2.6.m6.1.1.1.cmml" xref="S6.p2.6.m6.1.1">subscript</csymbol><apply id="S6.p2.6.m6.1.1.2.cmml" xref="S6.p2.6.m6.1.1.2"><ci id="S6.p2.6.m6.1.1.2.1.cmml" xref="S6.p2.6.m6.1.1.2.1">~</ci><ci id="S6.p2.6.m6.1.1.2.2.cmml" xref="S6.p2.6.m6.1.1.2.2">𝑝</ci></apply><cn id="S6.p2.6.m6.1.1.3.cmml" type="integer" xref="S6.p2.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.6.m6.1c">\tilde{p}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.6.m6.1d">over~ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> bound a region called the <span class="ltx_text ltx_font_italic" id="S6.p2.6.2">resonance zone</span>. (See also Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S5.F2" title="Figure 2 ‣ V Surface of Section and Discrete-Time Monte Carlo ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">2</span></a>.) Once a trajectory has escaped the resonance zone, it never returns. Physically, we interpret the resonance zone as the region where electrons are bound, or unionized.</p> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">It is important to study the topology of the manifolds to partition phase space. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a shows the minimum length of the stable and unstable manifolds needed to completely describe the topology of the infinitely long manifolds. The finite length of manifolds in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a is called a trellis. A trellis is the minimum set of manifolds which contains all the topological information required to construct the partition <cite class="ltx_cite ltx_citemacro_cite">Mitchell (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib20" title="">2012</a>)</cite>. The trellis can be used to determine a set of partition domains, or rectangles, and the symbolic dynamics between them. This can be done in a variety of ways <cite class="ltx_cite ltx_citemacro_cite">Gonzalez and Jung (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib27" title="">2014</a>)</cite>, including the method of homotopic lobe dynamics (HLD) <cite class="ltx_cite ltx_citemacro_cite">Mitchell (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib20" title="">2012</a>)</cite>. Here, however, we shall give an intuitive description of the partitioning and symbolic dynamics without the need for the full machinery of HLD. We first identify three topological rectangles formed by the trellis and shown shaded in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a. One can show that all trajectories that never leave the resonance zone in forward or backward time must lie within these three rectangles. The forward iterate of each of these rectangles is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>b, using the same coloring as in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a. Note that each iterated rectangle is stretched in length such that it passes through each of the original rectangles. We can develop a symbolic dynamics for this process by labeling the original rectangles ‘0’, ‘1’, ‘2’, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a. The iterated rectangles in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>b imply that there is an allowed transition from any of the three symbols to any of the three symbols, as shown by the transition graph in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a. This transition graph defines a Markov process, known as a full shift on three symbols <cite class="ltx_cite ltx_citemacro_cite">Jung, Lipp, and Seligman (<a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib28" title="">1999</a>)</cite>, that can be used to label all periodic orbits of the system.</p> </div> <figure class="ltx_figure" id="S6.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="501" id="S6.F3.g1" src="extracted/6294607/combinedTrellisE1.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S6.F3.6.3.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S6.F3.4.2" style="font-size:90%;"> Trellis at E = 1. 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id="S6.F3.4.2.m2.2.2.2.2.cmml" xref="S6.F3.4.2.m2.2.2.2.2">𝑝</ci><cn id="S6.F3.4.2.m2.2.2.2.3.cmml" type="integer" xref="S6.F3.4.2.m2.2.2.2.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.4.2.m2.4d">p_{0},\tilde{p_{0}},p_{1},\tilde{p_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.F3.4.2.m2.4e">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>, and labeled partition rectangles ‘0’, ‘1’, and ‘2’. Transition graph for transport between the rectangles shown in the bottom right. (b) The forward iterates of the partition rectangles from (a). (c) The trellis plotted with its backward iterate in red. Refined partition rectangles are colored and labeled. (d) The trellis and its backward iterate with all discrete periodic orbits up to period 6 plotted. Notice they are contained entirely within the refined partition rectangles from (c).</span></figcaption> </figure> <div class="ltx_para" id="S6.p4"> <p class="ltx_p" id="S6.p4.1">From the symbolic dynamics, the symbolic itinerary of every periodic orbit up to a given period can be written. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.T1" title="Table 1 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">1</span></a> shows the number of periodic orbits for a given length itinerary. For a faithful symbolic representation, as we have here, we can be sure that this method captures every orbit up to the given period. For each periodic itinerary, we can find the associated periodic orbit as follows. First, the symbolic itinerary labels a sequence of partition rectangles. Next, a single representative point from each rectangle is used as an initial approximation to the periodic orbit. Finally, this approximation is used as an initial condition in a multi-point shooting Newton’s method solver to quickly converge to the periodic orbit. For this approach to work, the initial guess must be sufficiently close to the true orbit for it to converge. In truth, the partition rectangles shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a are too large to provide sufficiently precise initial conditions to converge to the correct orbits. To address this issue we can refine any partition rectangle into smaller sub-rectangles with longer symbolic labels. The sub-rectangle labels correspond to both the current rectangle of a point and the next rectangle it will be mapped into.</p> </div> <div class="ltx_para" id="S6.p5"> <p class="ltx_p" id="S6.p5.1">To refine a partition, take a partition rectangle and map it forward one iterate. Note which partition rectangles it maps into. Take a region of overlap and map it backwards into the original rectangle. The resulting rectangle is a sub-rectangle corresponding to all trajectories that map from the starting rectangle to the second rectangle. Append a symbol to this new sub-rectangle’s label corresponding to the rectangle it gets mapped into. Each time this process is performed on a partition, exponentially more sub-rectangles will be generated to form a Cantor-like geometry. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>c shows the refined partition generated after one application of this method. Notice this has the same effect as mapping the stable manifold backward one iterate and labeling each region of overlap by the two rectangles that overlap. For any given orbit itinerary, use the center of each refined rectangle that corresponds to a substring of the orbit itinerary. For example, consider refining the ‘0’ rectangle. It maps to itself and each other rectangle, so refining it will give us three new sub-rectangles labeled ‘00’, ‘01’, and ‘02’ corresponding to each of the possible future locations of points originally within the ‘0’ rectangle. Thus, to find the periodic orbit with the itinerary ‘10’ choose the centers of the sub-rectangles labeled ‘01’ and ‘10’ to use as an initial guess. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>c shows the partition used to compute all periodic orbits up to <math alttext="n_{p}=10" class="ltx_Math" display="inline" id="S6.p5.1.m1.1"><semantics id="S6.p5.1.m1.1a"><mrow id="S6.p5.1.m1.1.1" xref="S6.p5.1.m1.1.1.cmml"><msub id="S6.p5.1.m1.1.1.2" xref="S6.p5.1.m1.1.1.2.cmml"><mi id="S6.p5.1.m1.1.1.2.2" xref="S6.p5.1.m1.1.1.2.2.cmml">n</mi><mi id="S6.p5.1.m1.1.1.2.3" xref="S6.p5.1.m1.1.1.2.3.cmml">p</mi></msub><mo id="S6.p5.1.m1.1.1.1" xref="S6.p5.1.m1.1.1.1.cmml">=</mo><mn id="S6.p5.1.m1.1.1.3" xref="S6.p5.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p5.1.m1.1b"><apply id="S6.p5.1.m1.1.1.cmml" xref="S6.p5.1.m1.1.1"><eq id="S6.p5.1.m1.1.1.1.cmml" xref="S6.p5.1.m1.1.1.1"></eq><apply id="S6.p5.1.m1.1.1.2.cmml" xref="S6.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.p5.1.m1.1.1.2.1.cmml" xref="S6.p5.1.m1.1.1.2">subscript</csymbol><ci id="S6.p5.1.m1.1.1.2.2.cmml" xref="S6.p5.1.m1.1.1.2.2">𝑛</ci><ci id="S6.p5.1.m1.1.1.2.3.cmml" xref="S6.p5.1.m1.1.1.2.3">𝑝</ci></apply><cn id="S6.p5.1.m1.1.1.3.cmml" type="integer" xref="S6.p5.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p5.1.m1.1c">n_{p}=10</annotation><annotation encoding="application/x-llamapun" id="S6.p5.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 10</annotation></semantics></math>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>d shows all discrete orbits up to period six plotted with the trellis.</p> </div> <div class="ltx_para" id="S6.p6"> <p class="ltx_p" id="S6.p6.5">To obtain the continuous-time trajectory in the full <math alttext="uvp_{u}p_{v}" class="ltx_Math" display="inline" id="S6.p6.1.m1.1"><semantics id="S6.p6.1.m1.1a"><mrow id="S6.p6.1.m1.1.1" xref="S6.p6.1.m1.1.1.cmml"><mi id="S6.p6.1.m1.1.1.2" xref="S6.p6.1.m1.1.1.2.cmml">u</mi><mo id="S6.p6.1.m1.1.1.1" xref="S6.p6.1.m1.1.1.1.cmml">⁢</mo><mi id="S6.p6.1.m1.1.1.3" xref="S6.p6.1.m1.1.1.3.cmml">v</mi><mo id="S6.p6.1.m1.1.1.1a" xref="S6.p6.1.m1.1.1.1.cmml">⁢</mo><msub id="S6.p6.1.m1.1.1.4" xref="S6.p6.1.m1.1.1.4.cmml"><mi id="S6.p6.1.m1.1.1.4.2" xref="S6.p6.1.m1.1.1.4.2.cmml">p</mi><mi id="S6.p6.1.m1.1.1.4.3" xref="S6.p6.1.m1.1.1.4.3.cmml">u</mi></msub><mo id="S6.p6.1.m1.1.1.1b" xref="S6.p6.1.m1.1.1.1.cmml">⁢</mo><msub id="S6.p6.1.m1.1.1.5" xref="S6.p6.1.m1.1.1.5.cmml"><mi id="S6.p6.1.m1.1.1.5.2" xref="S6.p6.1.m1.1.1.5.2.cmml">p</mi><mi id="S6.p6.1.m1.1.1.5.3" xref="S6.p6.1.m1.1.1.5.3.cmml">v</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.1.m1.1b"><apply id="S6.p6.1.m1.1.1.cmml" xref="S6.p6.1.m1.1.1"><times id="S6.p6.1.m1.1.1.1.cmml" xref="S6.p6.1.m1.1.1.1"></times><ci id="S6.p6.1.m1.1.1.2.cmml" xref="S6.p6.1.m1.1.1.2">𝑢</ci><ci id="S6.p6.1.m1.1.1.3.cmml" xref="S6.p6.1.m1.1.1.3">𝑣</ci><apply id="S6.p6.1.m1.1.1.4.cmml" xref="S6.p6.1.m1.1.1.4"><csymbol cd="ambiguous" id="S6.p6.1.m1.1.1.4.1.cmml" xref="S6.p6.1.m1.1.1.4">subscript</csymbol><ci id="S6.p6.1.m1.1.1.4.2.cmml" xref="S6.p6.1.m1.1.1.4.2">𝑝</ci><ci id="S6.p6.1.m1.1.1.4.3.cmml" xref="S6.p6.1.m1.1.1.4.3">𝑢</ci></apply><apply id="S6.p6.1.m1.1.1.5.cmml" xref="S6.p6.1.m1.1.1.5"><csymbol cd="ambiguous" id="S6.p6.1.m1.1.1.5.1.cmml" xref="S6.p6.1.m1.1.1.5">subscript</csymbol><ci id="S6.p6.1.m1.1.1.5.2.cmml" xref="S6.p6.1.m1.1.1.5.2">𝑝</ci><ci id="S6.p6.1.m1.1.1.5.3.cmml" xref="S6.p6.1.m1.1.1.5.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.1.m1.1c">uvp_{u}p_{v}</annotation><annotation encoding="application/x-llamapun" id="S6.p6.1.m1.1d">italic_u italic_v italic_p start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> phase space, integrate a discrete orbit forward through the full phase space. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F4" title="Figure 4 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">4</span></a> shows all continuous orbits up to period four translated into the original physical <math alttext="\rho z" class="ltx_Math" display="inline" id="S6.p6.2.m2.1"><semantics id="S6.p6.2.m2.1a"><mrow id="S6.p6.2.m2.1.1" xref="S6.p6.2.m2.1.1.cmml"><mi id="S6.p6.2.m2.1.1.2" xref="S6.p6.2.m2.1.1.2.cmml">ρ</mi><mo id="S6.p6.2.m2.1.1.1" xref="S6.p6.2.m2.1.1.1.cmml">⁢</mo><mi id="S6.p6.2.m2.1.1.3" xref="S6.p6.2.m2.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.2.m2.1b"><apply id="S6.p6.2.m2.1.1.cmml" xref="S6.p6.2.m2.1.1"><times id="S6.p6.2.m2.1.1.1.cmml" xref="S6.p6.2.m2.1.1.1"></times><ci id="S6.p6.2.m2.1.1.2.cmml" xref="S6.p6.2.m2.1.1.2">𝜌</ci><ci id="S6.p6.2.m2.1.1.3.cmml" xref="S6.p6.2.m2.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.2.m2.1c">\rho z</annotation><annotation encoding="application/x-llamapun" id="S6.p6.2.m2.1d">italic_ρ italic_z</annotation></semantics></math> configuration space. We have computed a complete set of all periodic orbits, both discrete and continuous, through discrete period <math alttext="n_{p}=10" class="ltx_Math" display="inline" id="S6.p6.3.m3.1"><semantics id="S6.p6.3.m3.1a"><mrow id="S6.p6.3.m3.1.1" xref="S6.p6.3.m3.1.1.cmml"><msub id="S6.p6.3.m3.1.1.2" xref="S6.p6.3.m3.1.1.2.cmml"><mi id="S6.p6.3.m3.1.1.2.2" xref="S6.p6.3.m3.1.1.2.2.cmml">n</mi><mi id="S6.p6.3.m3.1.1.2.3" xref="S6.p6.3.m3.1.1.2.3.cmml">p</mi></msub><mo id="S6.p6.3.m3.1.1.1" xref="S6.p6.3.m3.1.1.1.cmml">=</mo><mn id="S6.p6.3.m3.1.1.3" xref="S6.p6.3.m3.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.3.m3.1b"><apply id="S6.p6.3.m3.1.1.cmml" xref="S6.p6.3.m3.1.1"><eq id="S6.p6.3.m3.1.1.1.cmml" xref="S6.p6.3.m3.1.1.1"></eq><apply id="S6.p6.3.m3.1.1.2.cmml" xref="S6.p6.3.m3.1.1.2"><csymbol cd="ambiguous" id="S6.p6.3.m3.1.1.2.1.cmml" xref="S6.p6.3.m3.1.1.2">subscript</csymbol><ci id="S6.p6.3.m3.1.1.2.2.cmml" xref="S6.p6.3.m3.1.1.2.2">𝑛</ci><ci id="S6.p6.3.m3.1.1.2.3.cmml" xref="S6.p6.3.m3.1.1.2.3">𝑝</ci></apply><cn id="S6.p6.3.m3.1.1.3.cmml" type="integer" xref="S6.p6.3.m3.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.3.m3.1c">n_{p}=10</annotation><annotation encoding="application/x-llamapun" id="S6.p6.3.m3.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 10</annotation></semantics></math> for use below. We choose <math alttext="n_{p}=10" class="ltx_Math" display="inline" id="S6.p6.4.m4.1"><semantics id="S6.p6.4.m4.1a"><mrow id="S6.p6.4.m4.1.1" xref="S6.p6.4.m4.1.1.cmml"><msub id="S6.p6.4.m4.1.1.2" xref="S6.p6.4.m4.1.1.2.cmml"><mi id="S6.p6.4.m4.1.1.2.2" xref="S6.p6.4.m4.1.1.2.2.cmml">n</mi><mi id="S6.p6.4.m4.1.1.2.3" xref="S6.p6.4.m4.1.1.2.3.cmml">p</mi></msub><mo id="S6.p6.4.m4.1.1.1" xref="S6.p6.4.m4.1.1.1.cmml">=</mo><mn id="S6.p6.4.m4.1.1.3" xref="S6.p6.4.m4.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.4.m4.1b"><apply id="S6.p6.4.m4.1.1.cmml" xref="S6.p6.4.m4.1.1"><eq id="S6.p6.4.m4.1.1.1.cmml" xref="S6.p6.4.m4.1.1.1"></eq><apply id="S6.p6.4.m4.1.1.2.cmml" xref="S6.p6.4.m4.1.1.2"><csymbol cd="ambiguous" id="S6.p6.4.m4.1.1.2.1.cmml" xref="S6.p6.4.m4.1.1.2">subscript</csymbol><ci id="S6.p6.4.m4.1.1.2.2.cmml" xref="S6.p6.4.m4.1.1.2.2">𝑛</ci><ci id="S6.p6.4.m4.1.1.2.3.cmml" xref="S6.p6.4.m4.1.1.2.3">𝑝</ci></apply><cn id="S6.p6.4.m4.1.1.3.cmml" type="integer" xref="S6.p6.4.m4.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.4.m4.1c">n_{p}=10</annotation><annotation encoding="application/x-llamapun" id="S6.p6.4.m4.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 10</annotation></semantics></math> because we only need to refine the partition rectangles once and there is already a reasonably large number of orbits, <math alttext="\sim 10^{4}" class="ltx_Math" display="inline" id="S6.p6.5.m5.1"><semantics id="S6.p6.5.m5.1a"><mrow id="S6.p6.5.m5.1.1" xref="S6.p6.5.m5.1.1.cmml"><mi id="S6.p6.5.m5.1.1.2" xref="S6.p6.5.m5.1.1.2.cmml"></mi><mo id="S6.p6.5.m5.1.1.1" xref="S6.p6.5.m5.1.1.1.cmml">∼</mo><msup id="S6.p6.5.m5.1.1.3" xref="S6.p6.5.m5.1.1.3.cmml"><mn id="S6.p6.5.m5.1.1.3.2" xref="S6.p6.5.m5.1.1.3.2.cmml">10</mn><mn id="S6.p6.5.m5.1.1.3.3" xref="S6.p6.5.m5.1.1.3.3.cmml">4</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.5.m5.1b"><apply id="S6.p6.5.m5.1.1.cmml" xref="S6.p6.5.m5.1.1"><csymbol cd="latexml" id="S6.p6.5.m5.1.1.1.cmml" xref="S6.p6.5.m5.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S6.p6.5.m5.1.1.2.cmml" xref="S6.p6.5.m5.1.1.2">absent</csymbol><apply id="S6.p6.5.m5.1.1.3.cmml" xref="S6.p6.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.p6.5.m5.1.1.3.1.cmml" xref="S6.p6.5.m5.1.1.3">superscript</csymbol><cn id="S6.p6.5.m5.1.1.3.2.cmml" type="integer" xref="S6.p6.5.m5.1.1.3.2">10</cn><cn id="S6.p6.5.m5.1.1.3.3.cmml" type="integer" xref="S6.p6.5.m5.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.5.m5.1c">\sim 10^{4}</annotation><annotation encoding="application/x-llamapun" id="S6.p6.5.m5.1d">∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.p7"> <p class="ltx_p" id="S6.p7.1">In the set of continuous periodic orbits there are two symmetries that arise from symmetries of the underlying trellis. The first is time reversal symmetry corresponding to orbits that retrace themselves. The second is reflection symmetry about the vertical axis that corresponds to exchanging the ‘0’ and ‘2’ symbols in an orbits’ symbolic itinerary. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F4" title="Figure 4 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">4</span></a> shows continuous orbits up to period four colored by their invariance under these symmetry transformations. Some orbits exhibit invariance under the pure symmetry operations, while others are invariant under both separately, the composition of the two, or neither.</p> </div> <figure class="ltx_figure" id="S6.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="337" id="S6.F4.g1" src="extracted/6294607/periodicOrbitsUpTo4.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S6.F4.10.5.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S6.F4.8.4" style="font-size:90%;">All continuous orbits with itinerary lengths of one through four at <math alttext="E=1" class="ltx_Math" display="inline" id="S6.F4.5.1.m1.1"><semantics id="S6.F4.5.1.m1.1b"><mrow id="S6.F4.5.1.m1.1.1" xref="S6.F4.5.1.m1.1.1.cmml"><mi id="S6.F4.5.1.m1.1.1.2" xref="S6.F4.5.1.m1.1.1.2.cmml">E</mi><mo id="S6.F4.5.1.m1.1.1.1" xref="S6.F4.5.1.m1.1.1.1.cmml">=</mo><mn id="S6.F4.5.1.m1.1.1.3" xref="S6.F4.5.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F4.5.1.m1.1c"><apply id="S6.F4.5.1.m1.1.1.cmml" xref="S6.F4.5.1.m1.1.1"><eq id="S6.F4.5.1.m1.1.1.1.cmml" xref="S6.F4.5.1.m1.1.1.1"></eq><ci id="S6.F4.5.1.m1.1.1.2.cmml" xref="S6.F4.5.1.m1.1.1.2">𝐸</ci><cn id="S6.F4.5.1.m1.1.1.3.cmml" type="integer" xref="S6.F4.5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.5.1.m1.1d">E=1</annotation><annotation encoding="application/x-llamapun" id="S6.F4.5.1.m1.1e">italic_E = 1</annotation></semantics></math> and <math alttext="B=3.5" class="ltx_Math" display="inline" id="S6.F4.6.2.m2.1"><semantics id="S6.F4.6.2.m2.1b"><mrow id="S6.F4.6.2.m2.1.1" xref="S6.F4.6.2.m2.1.1.cmml"><mi id="S6.F4.6.2.m2.1.1.2" xref="S6.F4.6.2.m2.1.1.2.cmml">B</mi><mo id="S6.F4.6.2.m2.1.1.1" xref="S6.F4.6.2.m2.1.1.1.cmml">=</mo><mn id="S6.F4.6.2.m2.1.1.3" xref="S6.F4.6.2.m2.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F4.6.2.m2.1c"><apply id="S6.F4.6.2.m2.1.1.cmml" xref="S6.F4.6.2.m2.1.1"><eq id="S6.F4.6.2.m2.1.1.1.cmml" xref="S6.F4.6.2.m2.1.1.1"></eq><ci id="S6.F4.6.2.m2.1.1.2.cmml" xref="S6.F4.6.2.m2.1.1.2">𝐵</ci><cn id="S6.F4.6.2.m2.1.1.3.cmml" type="float" xref="S6.F4.6.2.m2.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.6.2.m2.1d">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S6.F4.6.2.m2.1e">italic_B = 3.5</annotation></semantics></math> plotted in the <math alttext="\rho z" class="ltx_Math" display="inline" id="S6.F4.7.3.m3.1"><semantics id="S6.F4.7.3.m3.1b"><mrow id="S6.F4.7.3.m3.1.1" xref="S6.F4.7.3.m3.1.1.cmml"><mi id="S6.F4.7.3.m3.1.1.2" xref="S6.F4.7.3.m3.1.1.2.cmml">ρ</mi><mo id="S6.F4.7.3.m3.1.1.1" xref="S6.F4.7.3.m3.1.1.1.cmml">⁢</mo><mi id="S6.F4.7.3.m3.1.1.3" xref="S6.F4.7.3.m3.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.F4.7.3.m3.1c"><apply id="S6.F4.7.3.m3.1.1.cmml" xref="S6.F4.7.3.m3.1.1"><times id="S6.F4.7.3.m3.1.1.1.cmml" xref="S6.F4.7.3.m3.1.1.1"></times><ci id="S6.F4.7.3.m3.1.1.2.cmml" xref="S6.F4.7.3.m3.1.1.2">𝜌</ci><ci id="S6.F4.7.3.m3.1.1.3.cmml" xref="S6.F4.7.3.m3.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.7.3.m3.1d">\rho z</annotation><annotation encoding="application/x-llamapun" id="S6.F4.7.3.m3.1e">italic_ρ italic_z</annotation></semantics></math> configuration space. The symbolic itinerary is noted above each orbit. Notice that the itinerary length does not precisely determine the length of the continuous orbit. Orbits highlighted in blue are invariant under time reversal. Orbits highlighted in red are invariant under both time reversal and the reflection ’0’<math alttext="\rightarrow" class="ltx_Math" display="inline" id="S6.F4.8.4.m4.1"><semantics id="S6.F4.8.4.m4.1b"><mo id="S6.F4.8.4.m4.1.1" stretchy="false" xref="S6.F4.8.4.m4.1.1.cmml">→</mo><annotation-xml encoding="MathML-Content" id="S6.F4.8.4.m4.1c"><ci id="S6.F4.8.4.m4.1.1.cmml" xref="S6.F4.8.4.m4.1.1">→</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.8.4.m4.1d">\rightarrow</annotation><annotation encoding="application/x-llamapun" id="S6.F4.8.4.m4.1e">→</annotation></semantics></math>’2’ separately. Orbits highlighted in yellow are invariant under the composition of time reversal and reflection. Orbits highlighted in green are invariant under neither time reversal nor reflection. Arrows indicate the direction of the orbit when the trajectory does not retrace itself.</span></figcaption> </figure> <figure class="ltx_table" id="S6.T1"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S6.T1.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S6.T1.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_r ltx_border_t" id="S6.T1.1.1.1">Period (<math alttext="n_{p}" class="ltx_Math" display="inline" id="S6.T1.1.1.1.m1.1"><semantics id="S6.T1.1.1.1.m1.1a"><msub id="S6.T1.1.1.1.m1.1.1" xref="S6.T1.1.1.1.m1.1.1.cmml"><mi id="S6.T1.1.1.1.m1.1.1.2" xref="S6.T1.1.1.1.m1.1.1.2.cmml">n</mi><mi id="S6.T1.1.1.1.m1.1.1.3" xref="S6.T1.1.1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S6.T1.1.1.1.m1.1b"><apply id="S6.T1.1.1.1.m1.1.1.cmml" xref="S6.T1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.T1.1.1.1.m1.1.1.1.cmml" xref="S6.T1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.T1.1.1.1.m1.1.1.2.cmml" xref="S6.T1.1.1.1.m1.1.1.2">𝑛</ci><ci id="S6.T1.1.1.1.m1.1.1.3.cmml" xref="S6.T1.1.1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.1.1.1.m1.1c">n_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.T1.1.1.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S6.T1.2.2.2"># Orbits <math alttext="E=1" class="ltx_Math" display="inline" id="S6.T1.2.2.2.m1.1"><semantics id="S6.T1.2.2.2.m1.1a"><mrow id="S6.T1.2.2.2.m1.1.1" xref="S6.T1.2.2.2.m1.1.1.cmml"><mi id="S6.T1.2.2.2.m1.1.1.2" xref="S6.T1.2.2.2.m1.1.1.2.cmml">E</mi><mo id="S6.T1.2.2.2.m1.1.1.1" xref="S6.T1.2.2.2.m1.1.1.1.cmml">=</mo><mn id="S6.T1.2.2.2.m1.1.1.3" xref="S6.T1.2.2.2.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.T1.2.2.2.m1.1b"><apply id="S6.T1.2.2.2.m1.1.1.cmml" xref="S6.T1.2.2.2.m1.1.1"><eq id="S6.T1.2.2.2.m1.1.1.1.cmml" xref="S6.T1.2.2.2.m1.1.1.1"></eq><ci id="S6.T1.2.2.2.m1.1.1.2.cmml" xref="S6.T1.2.2.2.m1.1.1.2">𝐸</ci><cn id="S6.T1.2.2.2.m1.1.1.3.cmml" type="integer" xref="S6.T1.2.2.2.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.2.2.2.m1.1c">E=1</annotation><annotation encoding="application/x-llamapun" id="S6.T1.2.2.2.m1.1d">italic_E = 1</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S6.T1.3.3.3"># Orbits <math alttext="E=0.285" class="ltx_Math" display="inline" id="S6.T1.3.3.3.m1.1"><semantics id="S6.T1.3.3.3.m1.1a"><mrow id="S6.T1.3.3.3.m1.1.1" xref="S6.T1.3.3.3.m1.1.1.cmml"><mi id="S6.T1.3.3.3.m1.1.1.2" xref="S6.T1.3.3.3.m1.1.1.2.cmml">E</mi><mo id="S6.T1.3.3.3.m1.1.1.1" xref="S6.T1.3.3.3.m1.1.1.1.cmml">=</mo><mn id="S6.T1.3.3.3.m1.1.1.3" xref="S6.T1.3.3.3.m1.1.1.3.cmml">0.285</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.T1.3.3.3.m1.1b"><apply id="S6.T1.3.3.3.m1.1.1.cmml" xref="S6.T1.3.3.3.m1.1.1"><eq id="S6.T1.3.3.3.m1.1.1.1.cmml" xref="S6.T1.3.3.3.m1.1.1.1"></eq><ci id="S6.T1.3.3.3.m1.1.1.2.cmml" xref="S6.T1.3.3.3.m1.1.1.2">𝐸</ci><cn id="S6.T1.3.3.3.m1.1.1.3.cmml" type="float" xref="S6.T1.3.3.3.m1.1.1.3">0.285</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.3.3.3.m1.1c">E=0.285</annotation><annotation encoding="application/x-llamapun" id="S6.T1.3.3.3.m1.1d">italic_E = 0.285</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S6.T1.3.4.1"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_tt" id="S6.T1.3.4.1.1">1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S6.T1.3.4.1.2">3</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S6.T1.3.4.1.3">3</td> </tr> <tr class="ltx_tr" id="S6.T1.3.5.2"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.5.2.1">2</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.5.2.2">3</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.5.2.3">3</td> </tr> <tr class="ltx_tr" id="S6.T1.3.6.3"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.6.3.1">3</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.6.3.2">8</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.6.3.3">4</td> </tr> <tr class="ltx_tr" id="S6.T1.3.7.4"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.7.4.1">4</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.7.4.2">18</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.7.4.3">14</td> </tr> <tr class="ltx_tr" id="S6.T1.3.8.5"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.8.5.1">5</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.8.5.2">48</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.8.5.3">44</td> </tr> <tr class="ltx_tr" id="S6.T1.3.9.6"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.9.6.1">6</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.9.6.2">116</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.9.6.3">110</td> </tr> <tr class="ltx_tr" id="S6.T1.3.10.7"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.10.7.1">7</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.10.7.2">312</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.10.7.3">284</td> </tr> <tr class="ltx_tr" id="S6.T1.3.11.8"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.11.8.1">8</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.11.8.2">810</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.11.8.3">716</td> </tr> <tr class="ltx_tr" id="S6.T1.3.12.9"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S6.T1.3.12.9.1">9</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.12.9.2">2184</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S6.T1.3.12.9.3">1888</td> </tr> <tr class="ltx_tr" id="S6.T1.3.13.10"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r" id="S6.T1.3.13.10.1">10</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S6.T1.3.13.10.2">5880</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S6.T1.3.13.10.3">4998</td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S6.T1.13.5.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S6.T1.11.4" style="font-size:90%;">Number of periodic orbits at <math alttext="B=3.5" class="ltx_Math" display="inline" id="S6.T1.8.1.m1.1"><semantics id="S6.T1.8.1.m1.1b"><mrow id="S6.T1.8.1.m1.1.1" xref="S6.T1.8.1.m1.1.1.cmml"><mi id="S6.T1.8.1.m1.1.1.2" xref="S6.T1.8.1.m1.1.1.2.cmml">B</mi><mo id="S6.T1.8.1.m1.1.1.1" xref="S6.T1.8.1.m1.1.1.1.cmml">=</mo><mn id="S6.T1.8.1.m1.1.1.3" xref="S6.T1.8.1.m1.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.T1.8.1.m1.1c"><apply id="S6.T1.8.1.m1.1.1.cmml" xref="S6.T1.8.1.m1.1.1"><eq id="S6.T1.8.1.m1.1.1.1.cmml" xref="S6.T1.8.1.m1.1.1.1"></eq><ci id="S6.T1.8.1.m1.1.1.2.cmml" xref="S6.T1.8.1.m1.1.1.2">𝐵</ci><cn id="S6.T1.8.1.m1.1.1.3.cmml" type="float" xref="S6.T1.8.1.m1.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.8.1.m1.1d">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S6.T1.8.1.m1.1e">italic_B = 3.5</annotation></semantics></math> for each discrete period up to <math alttext="n_{p}=10" class="ltx_Math" display="inline" id="S6.T1.9.2.m2.1"><semantics id="S6.T1.9.2.m2.1b"><mrow id="S6.T1.9.2.m2.1.1" xref="S6.T1.9.2.m2.1.1.cmml"><msub id="S6.T1.9.2.m2.1.1.2" xref="S6.T1.9.2.m2.1.1.2.cmml"><mi id="S6.T1.9.2.m2.1.1.2.2" xref="S6.T1.9.2.m2.1.1.2.2.cmml">n</mi><mi id="S6.T1.9.2.m2.1.1.2.3" xref="S6.T1.9.2.m2.1.1.2.3.cmml">p</mi></msub><mo id="S6.T1.9.2.m2.1.1.1" xref="S6.T1.9.2.m2.1.1.1.cmml">=</mo><mn id="S6.T1.9.2.m2.1.1.3" xref="S6.T1.9.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.T1.9.2.m2.1c"><apply id="S6.T1.9.2.m2.1.1.cmml" xref="S6.T1.9.2.m2.1.1"><eq id="S6.T1.9.2.m2.1.1.1.cmml" xref="S6.T1.9.2.m2.1.1.1"></eq><apply id="S6.T1.9.2.m2.1.1.2.cmml" xref="S6.T1.9.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.T1.9.2.m2.1.1.2.1.cmml" xref="S6.T1.9.2.m2.1.1.2">subscript</csymbol><ci id="S6.T1.9.2.m2.1.1.2.2.cmml" xref="S6.T1.9.2.m2.1.1.2.2">𝑛</ci><ci id="S6.T1.9.2.m2.1.1.2.3.cmml" xref="S6.T1.9.2.m2.1.1.2.3">𝑝</ci></apply><cn id="S6.T1.9.2.m2.1.1.3.cmml" type="integer" xref="S6.T1.9.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.9.2.m2.1d">n_{p}=10</annotation><annotation encoding="application/x-llamapun" id="S6.T1.9.2.m2.1e">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 10</annotation></semantics></math>. Values shown for <math alttext="E=1.0" class="ltx_Math" display="inline" id="S6.T1.10.3.m3.1"><semantics id="S6.T1.10.3.m3.1b"><mrow id="S6.T1.10.3.m3.1.1" xref="S6.T1.10.3.m3.1.1.cmml"><mi id="S6.T1.10.3.m3.1.1.2" xref="S6.T1.10.3.m3.1.1.2.cmml">E</mi><mo id="S6.T1.10.3.m3.1.1.1" xref="S6.T1.10.3.m3.1.1.1.cmml">=</mo><mn id="S6.T1.10.3.m3.1.1.3" xref="S6.T1.10.3.m3.1.1.3.cmml">1.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.T1.10.3.m3.1c"><apply id="S6.T1.10.3.m3.1.1.cmml" xref="S6.T1.10.3.m3.1.1"><eq id="S6.T1.10.3.m3.1.1.1.cmml" xref="S6.T1.10.3.m3.1.1.1"></eq><ci id="S6.T1.10.3.m3.1.1.2.cmml" xref="S6.T1.10.3.m3.1.1.2">𝐸</ci><cn id="S6.T1.10.3.m3.1.1.3.cmml" type="float" xref="S6.T1.10.3.m3.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.10.3.m3.1d">E=1.0</annotation><annotation encoding="application/x-llamapun" id="S6.T1.10.3.m3.1e">italic_E = 1.0</annotation></semantics></math> and <math alttext="E=0.285" class="ltx_Math" display="inline" id="S6.T1.11.4.m4.1"><semantics id="S6.T1.11.4.m4.1b"><mrow id="S6.T1.11.4.m4.1.1" xref="S6.T1.11.4.m4.1.1.cmml"><mi id="S6.T1.11.4.m4.1.1.2" xref="S6.T1.11.4.m4.1.1.2.cmml">E</mi><mo id="S6.T1.11.4.m4.1.1.1" xref="S6.T1.11.4.m4.1.1.1.cmml">=</mo><mn id="S6.T1.11.4.m4.1.1.3" xref="S6.T1.11.4.m4.1.1.3.cmml">0.285</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.T1.11.4.m4.1c"><apply id="S6.T1.11.4.m4.1.1.cmml" xref="S6.T1.11.4.m4.1.1"><eq id="S6.T1.11.4.m4.1.1.1.cmml" xref="S6.T1.11.4.m4.1.1.1"></eq><ci id="S6.T1.11.4.m4.1.1.2.cmml" xref="S6.T1.11.4.m4.1.1.2">𝐸</ci><cn id="S6.T1.11.4.m4.1.1.3.cmml" type="float" xref="S6.T1.11.4.m4.1.1.3">0.285</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.11.4.m4.1d">E=0.285</annotation><annotation encoding="application/x-llamapun" id="S6.T1.11.4.m4.1e">italic_E = 0.285</annotation></semantics></math>, corresponding to the upper and lower plateau, respectively. The values were computed directly from the transition graphs in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F7" title="Figure 7 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">7</span></a>b, and they match the number of numerically computed periodic orbits exactly.</span></figcaption> </figure> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VII </span>Escape Rate from Periodic Orbits</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.6">Now that we have computed a full set of periodic orbits up to <math alttext="n_{p}=10" class="ltx_Math" display="inline" id="S7.p1.1.m1.1"><semantics id="S7.p1.1.m1.1a"><mrow id="S7.p1.1.m1.1.1" xref="S7.p1.1.m1.1.1.cmml"><msub id="S7.p1.1.m1.1.1.2" xref="S7.p1.1.m1.1.1.2.cmml"><mi id="S7.p1.1.m1.1.1.2.2" xref="S7.p1.1.m1.1.1.2.2.cmml">n</mi><mi id="S7.p1.1.m1.1.1.2.3" xref="S7.p1.1.m1.1.1.2.3.cmml">p</mi></msub><mo id="S7.p1.1.m1.1.1.1" xref="S7.p1.1.m1.1.1.1.cmml">=</mo><mn id="S7.p1.1.m1.1.1.3" xref="S7.p1.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.1.m1.1b"><apply id="S7.p1.1.m1.1.1.cmml" xref="S7.p1.1.m1.1.1"><eq id="S7.p1.1.m1.1.1.1.cmml" xref="S7.p1.1.m1.1.1.1"></eq><apply id="S7.p1.1.m1.1.1.2.cmml" xref="S7.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S7.p1.1.m1.1.1.2.1.cmml" xref="S7.p1.1.m1.1.1.2">subscript</csymbol><ci id="S7.p1.1.m1.1.1.2.2.cmml" xref="S7.p1.1.m1.1.1.2.2">𝑛</ci><ci id="S7.p1.1.m1.1.1.2.3.cmml" xref="S7.p1.1.m1.1.1.2.3">𝑝</ci></apply><cn id="S7.p1.1.m1.1.1.3.cmml" type="integer" xref="S7.p1.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.1.m1.1c">n_{p}=10</annotation><annotation encoding="application/x-llamapun" id="S7.p1.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 10</annotation></semantics></math> at <math alttext="B=3.5,E=1.0" class="ltx_Math" display="inline" id="S7.p1.2.m2.2"><semantics id="S7.p1.2.m2.2a"><mrow id="S7.p1.2.m2.2.2.2" xref="S7.p1.2.m2.2.2.3.cmml"><mrow id="S7.p1.2.m2.1.1.1.1" xref="S7.p1.2.m2.1.1.1.1.cmml"><mi id="S7.p1.2.m2.1.1.1.1.2" xref="S7.p1.2.m2.1.1.1.1.2.cmml">B</mi><mo id="S7.p1.2.m2.1.1.1.1.1" xref="S7.p1.2.m2.1.1.1.1.1.cmml">=</mo><mn id="S7.p1.2.m2.1.1.1.1.3" xref="S7.p1.2.m2.1.1.1.1.3.cmml">3.5</mn></mrow><mo id="S7.p1.2.m2.2.2.2.3" xref="S7.p1.2.m2.2.2.3a.cmml">,</mo><mrow id="S7.p1.2.m2.2.2.2.2" xref="S7.p1.2.m2.2.2.2.2.cmml"><mi id="S7.p1.2.m2.2.2.2.2.2" xref="S7.p1.2.m2.2.2.2.2.2.cmml">E</mi><mo id="S7.p1.2.m2.2.2.2.2.1" xref="S7.p1.2.m2.2.2.2.2.1.cmml">=</mo><mn id="S7.p1.2.m2.2.2.2.2.3" xref="S7.p1.2.m2.2.2.2.2.3.cmml">1.0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.2.m2.2b"><apply id="S7.p1.2.m2.2.2.3.cmml" xref="S7.p1.2.m2.2.2.2"><csymbol cd="ambiguous" id="S7.p1.2.m2.2.2.3a.cmml" xref="S7.p1.2.m2.2.2.2.3">formulae-sequence</csymbol><apply id="S7.p1.2.m2.1.1.1.1.cmml" xref="S7.p1.2.m2.1.1.1.1"><eq id="S7.p1.2.m2.1.1.1.1.1.cmml" xref="S7.p1.2.m2.1.1.1.1.1"></eq><ci id="S7.p1.2.m2.1.1.1.1.2.cmml" xref="S7.p1.2.m2.1.1.1.1.2">𝐵</ci><cn id="S7.p1.2.m2.1.1.1.1.3.cmml" type="float" xref="S7.p1.2.m2.1.1.1.1.3">3.5</cn></apply><apply id="S7.p1.2.m2.2.2.2.2.cmml" xref="S7.p1.2.m2.2.2.2.2"><eq id="S7.p1.2.m2.2.2.2.2.1.cmml" xref="S7.p1.2.m2.2.2.2.2.1"></eq><ci id="S7.p1.2.m2.2.2.2.2.2.cmml" xref="S7.p1.2.m2.2.2.2.2.2">𝐸</ci><cn id="S7.p1.2.m2.2.2.2.2.3.cmml" type="float" xref="S7.p1.2.m2.2.2.2.2.3">1.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.2.m2.2c">B=3.5,E=1.0</annotation><annotation encoding="application/x-llamapun" id="S7.p1.2.m2.2d">italic_B = 3.5 , italic_E = 1.0</annotation></semantics></math>, we can utilize roots of the spectral determinants to compute the escape rate. Starting with the discrete-time case, we compute the escape rate to be <math alttext="\gamma_{d}=0.860180\pm 2.1\times 10^{-5}" class="ltx_Math" display="inline" id="S7.p1.3.m3.1"><semantics id="S7.p1.3.m3.1a"><mrow id="S7.p1.3.m3.1.1" xref="S7.p1.3.m3.1.1.cmml"><msub id="S7.p1.3.m3.1.1.2" xref="S7.p1.3.m3.1.1.2.cmml"><mi id="S7.p1.3.m3.1.1.2.2" xref="S7.p1.3.m3.1.1.2.2.cmml">γ</mi><mi id="S7.p1.3.m3.1.1.2.3" xref="S7.p1.3.m3.1.1.2.3.cmml">d</mi></msub><mo id="S7.p1.3.m3.1.1.1" xref="S7.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S7.p1.3.m3.1.1.3" xref="S7.p1.3.m3.1.1.3.cmml"><mn id="S7.p1.3.m3.1.1.3.2" xref="S7.p1.3.m3.1.1.3.2.cmml">0.860180</mn><mo id="S7.p1.3.m3.1.1.3.1" xref="S7.p1.3.m3.1.1.3.1.cmml">±</mo><mrow id="S7.p1.3.m3.1.1.3.3" xref="S7.p1.3.m3.1.1.3.3.cmml"><mn id="S7.p1.3.m3.1.1.3.3.2" xref="S7.p1.3.m3.1.1.3.3.2.cmml">2.1</mn><mo id="S7.p1.3.m3.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S7.p1.3.m3.1.1.3.3.1.cmml">×</mo><msup id="S7.p1.3.m3.1.1.3.3.3" xref="S7.p1.3.m3.1.1.3.3.3.cmml"><mn id="S7.p1.3.m3.1.1.3.3.3.2" xref="S7.p1.3.m3.1.1.3.3.3.2.cmml">10</mn><mrow id="S7.p1.3.m3.1.1.3.3.3.3" xref="S7.p1.3.m3.1.1.3.3.3.3.cmml"><mo id="S7.p1.3.m3.1.1.3.3.3.3a" xref="S7.p1.3.m3.1.1.3.3.3.3.cmml">−</mo><mn id="S7.p1.3.m3.1.1.3.3.3.3.2" xref="S7.p1.3.m3.1.1.3.3.3.3.2.cmml">5</mn></mrow></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.3.m3.1b"><apply id="S7.p1.3.m3.1.1.cmml" xref="S7.p1.3.m3.1.1"><eq id="S7.p1.3.m3.1.1.1.cmml" xref="S7.p1.3.m3.1.1.1"></eq><apply id="S7.p1.3.m3.1.1.2.cmml" xref="S7.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S7.p1.3.m3.1.1.2.1.cmml" xref="S7.p1.3.m3.1.1.2">subscript</csymbol><ci id="S7.p1.3.m3.1.1.2.2.cmml" xref="S7.p1.3.m3.1.1.2.2">𝛾</ci><ci id="S7.p1.3.m3.1.1.2.3.cmml" xref="S7.p1.3.m3.1.1.2.3">𝑑</ci></apply><apply id="S7.p1.3.m3.1.1.3.cmml" xref="S7.p1.3.m3.1.1.3"><csymbol cd="latexml" id="S7.p1.3.m3.1.1.3.1.cmml" xref="S7.p1.3.m3.1.1.3.1">plus-or-minus</csymbol><cn id="S7.p1.3.m3.1.1.3.2.cmml" type="float" xref="S7.p1.3.m3.1.1.3.2">0.860180</cn><apply id="S7.p1.3.m3.1.1.3.3.cmml" xref="S7.p1.3.m3.1.1.3.3"><times id="S7.p1.3.m3.1.1.3.3.1.cmml" xref="S7.p1.3.m3.1.1.3.3.1"></times><cn id="S7.p1.3.m3.1.1.3.3.2.cmml" type="float" xref="S7.p1.3.m3.1.1.3.3.2">2.1</cn><apply id="S7.p1.3.m3.1.1.3.3.3.cmml" xref="S7.p1.3.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S7.p1.3.m3.1.1.3.3.3.1.cmml" xref="S7.p1.3.m3.1.1.3.3.3">superscript</csymbol><cn id="S7.p1.3.m3.1.1.3.3.3.2.cmml" type="integer" xref="S7.p1.3.m3.1.1.3.3.3.2">10</cn><apply id="S7.p1.3.m3.1.1.3.3.3.3.cmml" xref="S7.p1.3.m3.1.1.3.3.3.3"><minus id="S7.p1.3.m3.1.1.3.3.3.3.1.cmml" xref="S7.p1.3.m3.1.1.3.3.3.3"></minus><cn id="S7.p1.3.m3.1.1.3.3.3.3.2.cmml" type="integer" xref="S7.p1.3.m3.1.1.3.3.3.3.2">5</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.3.m3.1c">\gamma_{d}=0.860180\pm 2.1\times 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S7.p1.3.m3.1d">italic_γ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0.860180 ± 2.1 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>. This agrees well with the value computed previously from Monte Carlo data <math alttext="\gamma_{d}=0.8456\pm 0.012" class="ltx_Math" display="inline" id="S7.p1.4.m4.1"><semantics id="S7.p1.4.m4.1a"><mrow id="S7.p1.4.m4.1.1" xref="S7.p1.4.m4.1.1.cmml"><msub id="S7.p1.4.m4.1.1.2" xref="S7.p1.4.m4.1.1.2.cmml"><mi id="S7.p1.4.m4.1.1.2.2" xref="S7.p1.4.m4.1.1.2.2.cmml">γ</mi><mi id="S7.p1.4.m4.1.1.2.3" xref="S7.p1.4.m4.1.1.2.3.cmml">d</mi></msub><mo id="S7.p1.4.m4.1.1.1" xref="S7.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S7.p1.4.m4.1.1.3" xref="S7.p1.4.m4.1.1.3.cmml"><mn id="S7.p1.4.m4.1.1.3.2" xref="S7.p1.4.m4.1.1.3.2.cmml">0.8456</mn><mo id="S7.p1.4.m4.1.1.3.1" xref="S7.p1.4.m4.1.1.3.1.cmml">±</mo><mn id="S7.p1.4.m4.1.1.3.3" xref="S7.p1.4.m4.1.1.3.3.cmml">0.012</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.4.m4.1b"><apply id="S7.p1.4.m4.1.1.cmml" xref="S7.p1.4.m4.1.1"><eq id="S7.p1.4.m4.1.1.1.cmml" xref="S7.p1.4.m4.1.1.1"></eq><apply id="S7.p1.4.m4.1.1.2.cmml" xref="S7.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S7.p1.4.m4.1.1.2.1.cmml" xref="S7.p1.4.m4.1.1.2">subscript</csymbol><ci id="S7.p1.4.m4.1.1.2.2.cmml" xref="S7.p1.4.m4.1.1.2.2">𝛾</ci><ci id="S7.p1.4.m4.1.1.2.3.cmml" xref="S7.p1.4.m4.1.1.2.3">𝑑</ci></apply><apply id="S7.p1.4.m4.1.1.3.cmml" xref="S7.p1.4.m4.1.1.3"><csymbol cd="latexml" id="S7.p1.4.m4.1.1.3.1.cmml" xref="S7.p1.4.m4.1.1.3.1">plus-or-minus</csymbol><cn id="S7.p1.4.m4.1.1.3.2.cmml" type="float" xref="S7.p1.4.m4.1.1.3.2">0.8456</cn><cn id="S7.p1.4.m4.1.1.3.3.cmml" type="float" xref="S7.p1.4.m4.1.1.3.3">0.012</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.4.m4.1c">\gamma_{d}=0.8456\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S7.p1.4.m4.1d">italic_γ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0.8456 ± 0.012</annotation></semantics></math>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>a shows the convergence of the discrete spectral determinant as a function of the highest period orbit used. The error is computed by taking the absolute difference between the escape rate computed with all orbits up to <math alttext="n_{p}=10" class="ltx_Math" display="inline" id="S7.p1.5.m5.1"><semantics id="S7.p1.5.m5.1a"><mrow id="S7.p1.5.m5.1.1" xref="S7.p1.5.m5.1.1.cmml"><msub id="S7.p1.5.m5.1.1.2" xref="S7.p1.5.m5.1.1.2.cmml"><mi id="S7.p1.5.m5.1.1.2.2" xref="S7.p1.5.m5.1.1.2.2.cmml">n</mi><mi id="S7.p1.5.m5.1.1.2.3" xref="S7.p1.5.m5.1.1.2.3.cmml">p</mi></msub><mo id="S7.p1.5.m5.1.1.1" xref="S7.p1.5.m5.1.1.1.cmml">=</mo><mn id="S7.p1.5.m5.1.1.3" xref="S7.p1.5.m5.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.5.m5.1b"><apply id="S7.p1.5.m5.1.1.cmml" xref="S7.p1.5.m5.1.1"><eq id="S7.p1.5.m5.1.1.1.cmml" xref="S7.p1.5.m5.1.1.1"></eq><apply id="S7.p1.5.m5.1.1.2.cmml" xref="S7.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S7.p1.5.m5.1.1.2.1.cmml" xref="S7.p1.5.m5.1.1.2">subscript</csymbol><ci id="S7.p1.5.m5.1.1.2.2.cmml" xref="S7.p1.5.m5.1.1.2.2">𝑛</ci><ci id="S7.p1.5.m5.1.1.2.3.cmml" xref="S7.p1.5.m5.1.1.2.3">𝑝</ci></apply><cn id="S7.p1.5.m5.1.1.3.cmml" type="integer" xref="S7.p1.5.m5.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.5.m5.1c">n_{p}=10</annotation><annotation encoding="application/x-llamapun" id="S7.p1.5.m5.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 10</annotation></semantics></math> and <math alttext="n_{p}=9" class="ltx_Math" display="inline" id="S7.p1.6.m6.1"><semantics id="S7.p1.6.m6.1a"><mrow id="S7.p1.6.m6.1.1" xref="S7.p1.6.m6.1.1.cmml"><msub id="S7.p1.6.m6.1.1.2" xref="S7.p1.6.m6.1.1.2.cmml"><mi id="S7.p1.6.m6.1.1.2.2" xref="S7.p1.6.m6.1.1.2.2.cmml">n</mi><mi id="S7.p1.6.m6.1.1.2.3" xref="S7.p1.6.m6.1.1.2.3.cmml">p</mi></msub><mo id="S7.p1.6.m6.1.1.1" xref="S7.p1.6.m6.1.1.1.cmml">=</mo><mn id="S7.p1.6.m6.1.1.3" xref="S7.p1.6.m6.1.1.3.cmml">9</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.6.m6.1b"><apply id="S7.p1.6.m6.1.1.cmml" xref="S7.p1.6.m6.1.1"><eq id="S7.p1.6.m6.1.1.1.cmml" xref="S7.p1.6.m6.1.1.1"></eq><apply id="S7.p1.6.m6.1.1.2.cmml" xref="S7.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S7.p1.6.m6.1.1.2.1.cmml" xref="S7.p1.6.m6.1.1.2">subscript</csymbol><ci id="S7.p1.6.m6.1.1.2.2.cmml" xref="S7.p1.6.m6.1.1.2.2">𝑛</ci><ci id="S7.p1.6.m6.1.1.2.3.cmml" xref="S7.p1.6.m6.1.1.2.3">𝑝</ci></apply><cn id="S7.p1.6.m6.1.1.3.cmml" type="integer" xref="S7.p1.6.m6.1.1.3">9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.6.m6.1c">n_{p}=9</annotation><annotation encoding="application/x-llamapun" id="S7.p1.6.m6.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 9</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.p2"> <p class="ltx_p" id="S7.p2.2">For the continuous case, we compute the escape rate to be <math alttext="\gamma=0.3882\pm 5.1\times 10^{-3}" class="ltx_Math" display="inline" id="S7.p2.1.m1.1"><semantics id="S7.p2.1.m1.1a"><mrow id="S7.p2.1.m1.1.1" xref="S7.p2.1.m1.1.1.cmml"><mi id="S7.p2.1.m1.1.1.2" xref="S7.p2.1.m1.1.1.2.cmml">γ</mi><mo id="S7.p2.1.m1.1.1.1" xref="S7.p2.1.m1.1.1.1.cmml">=</mo><mrow id="S7.p2.1.m1.1.1.3" xref="S7.p2.1.m1.1.1.3.cmml"><mn id="S7.p2.1.m1.1.1.3.2" xref="S7.p2.1.m1.1.1.3.2.cmml">0.3882</mn><mo id="S7.p2.1.m1.1.1.3.1" xref="S7.p2.1.m1.1.1.3.1.cmml">±</mo><mrow id="S7.p2.1.m1.1.1.3.3" xref="S7.p2.1.m1.1.1.3.3.cmml"><mn id="S7.p2.1.m1.1.1.3.3.2" xref="S7.p2.1.m1.1.1.3.3.2.cmml">5.1</mn><mo id="S7.p2.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S7.p2.1.m1.1.1.3.3.1.cmml">×</mo><msup id="S7.p2.1.m1.1.1.3.3.3" xref="S7.p2.1.m1.1.1.3.3.3.cmml"><mn id="S7.p2.1.m1.1.1.3.3.3.2" xref="S7.p2.1.m1.1.1.3.3.3.2.cmml">10</mn><mrow id="S7.p2.1.m1.1.1.3.3.3.3" xref="S7.p2.1.m1.1.1.3.3.3.3.cmml"><mo id="S7.p2.1.m1.1.1.3.3.3.3a" xref="S7.p2.1.m1.1.1.3.3.3.3.cmml">−</mo><mn id="S7.p2.1.m1.1.1.3.3.3.3.2" xref="S7.p2.1.m1.1.1.3.3.3.3.2.cmml">3</mn></mrow></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.p2.1.m1.1b"><apply id="S7.p2.1.m1.1.1.cmml" xref="S7.p2.1.m1.1.1"><eq id="S7.p2.1.m1.1.1.1.cmml" xref="S7.p2.1.m1.1.1.1"></eq><ci id="S7.p2.1.m1.1.1.2.cmml" xref="S7.p2.1.m1.1.1.2">𝛾</ci><apply id="S7.p2.1.m1.1.1.3.cmml" xref="S7.p2.1.m1.1.1.3"><csymbol cd="latexml" id="S7.p2.1.m1.1.1.3.1.cmml" xref="S7.p2.1.m1.1.1.3.1">plus-or-minus</csymbol><cn id="S7.p2.1.m1.1.1.3.2.cmml" type="float" xref="S7.p2.1.m1.1.1.3.2">0.3882</cn><apply id="S7.p2.1.m1.1.1.3.3.cmml" xref="S7.p2.1.m1.1.1.3.3"><times id="S7.p2.1.m1.1.1.3.3.1.cmml" xref="S7.p2.1.m1.1.1.3.3.1"></times><cn id="S7.p2.1.m1.1.1.3.3.2.cmml" type="float" xref="S7.p2.1.m1.1.1.3.3.2">5.1</cn><apply id="S7.p2.1.m1.1.1.3.3.3.cmml" xref="S7.p2.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S7.p2.1.m1.1.1.3.3.3.1.cmml" xref="S7.p2.1.m1.1.1.3.3.3">superscript</csymbol><cn id="S7.p2.1.m1.1.1.3.3.3.2.cmml" type="integer" xref="S7.p2.1.m1.1.1.3.3.3.2">10</cn><apply id="S7.p2.1.m1.1.1.3.3.3.3.cmml" xref="S7.p2.1.m1.1.1.3.3.3.3"><minus id="S7.p2.1.m1.1.1.3.3.3.3.1.cmml" xref="S7.p2.1.m1.1.1.3.3.3.3"></minus><cn id="S7.p2.1.m1.1.1.3.3.3.3.2.cmml" type="integer" xref="S7.p2.1.m1.1.1.3.3.3.3.2">3</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.1.m1.1c">\gamma=0.3882\pm 5.1\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S7.p2.1.m1.1d">italic_γ = 0.3882 ± 5.1 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math>. The error is computed in the same way as the discrete case. While the discrete case has excellent agreement with the Monte Carlo simulation, the continuous case has not quite converged. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>b shows the convergence of the spectral determinant for the continuous case. Unlike the discrete case, which has clearly converged, the continuous case requires higher period orbits to reach convergence. The convergence can be dramatically improved by decreasing the value of <math alttext="E" class="ltx_Math" display="inline" id="S7.p2.2.m2.1"><semantics id="S7.p2.2.m2.1a"><mi id="S7.p2.2.m2.1.1" xref="S7.p2.2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S7.p2.2.m2.1b"><ci id="S7.p2.2.m2.1.1.cmml" xref="S7.p2.2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.2.m2.1c">E</annotation><annotation encoding="application/x-llamapun" id="S7.p2.2.m2.1d">italic_E</annotation></semantics></math>, which we discuss in detail below.</p> </div> <figure class="ltx_figure" id="S7.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="782" id="S7.F5.g1" src="extracted/6294607/decayByPeriodCombined.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S7.F5.6.3.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="S7.F5.4.2" style="font-size:90%;">Decay rate computed with an increasing set of orbits. At each point all orbits up to the ’maximum period’ are used to compute the decay rate. Discrete data is shown in (a), (c), and (e). Continuous data is shown in (b), (d), and (f). Each figure is computed at <math alttext="B=3.5" class="ltx_Math" display="inline" id="S7.F5.3.1.m1.1"><semantics id="S7.F5.3.1.m1.1b"><mrow id="S7.F5.3.1.m1.1.1" xref="S7.F5.3.1.m1.1.1.cmml"><mi id="S7.F5.3.1.m1.1.1.2" xref="S7.F5.3.1.m1.1.1.2.cmml">B</mi><mo id="S7.F5.3.1.m1.1.1.1" xref="S7.F5.3.1.m1.1.1.1.cmml">=</mo><mn id="S7.F5.3.1.m1.1.1.3" xref="S7.F5.3.1.m1.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.F5.3.1.m1.1c"><apply id="S7.F5.3.1.m1.1.1.cmml" xref="S7.F5.3.1.m1.1.1"><eq id="S7.F5.3.1.m1.1.1.1.cmml" xref="S7.F5.3.1.m1.1.1.1"></eq><ci id="S7.F5.3.1.m1.1.1.2.cmml" xref="S7.F5.3.1.m1.1.1.2">𝐵</ci><cn id="S7.F5.3.1.m1.1.1.3.cmml" type="float" xref="S7.F5.3.1.m1.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F5.3.1.m1.1d">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S7.F5.3.1.m1.1e">italic_B = 3.5</annotation></semantics></math> with a different values of <math alttext="E" class="ltx_Math" display="inline" id="S7.F5.4.2.m2.1"><semantics id="S7.F5.4.2.m2.1b"><mi id="S7.F5.4.2.m2.1.1" xref="S7.F5.4.2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S7.F5.4.2.m2.1c"><ci id="S7.F5.4.2.m2.1.1.cmml" xref="S7.F5.4.2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F5.4.2.m2.1d">E</annotation><annotation encoding="application/x-llamapun" id="S7.F5.4.2.m2.1e">italic_E</annotation></semantics></math> to show the difference in convergence between the regions. For the discrete case all parameter values have clearly converged. For the continuous case (b) and (d) have clearly converged while (f) has not yet converged. More orbits would be needed for convergence at that parameter value.</span></figcaption> </figure> </section> <section class="ltx_section" id="S8"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VIII </span>Locating Hyperbolic Plateaus</h2> <div class="ltx_para" id="S8.p1"> <p class="ltx_p" id="S8.p1.2">Generally, the symbolic dynamics change as system parameters, i.e. <math alttext="E" class="ltx_Math" display="inline" id="S8.p1.1.m1.1"><semantics id="S8.p1.1.m1.1a"><mi id="S8.p1.1.m1.1.1" xref="S8.p1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S8.p1.1.m1.1b"><ci id="S8.p1.1.m1.1.1.cmml" xref="S8.p1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S8.p1.1.m1.1d">italic_E</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S8.p1.2.m2.1"><semantics id="S8.p1.2.m2.1a"><mi id="S8.p1.2.m2.1.1" xref="S8.p1.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S8.p1.2.m2.1b"><ci id="S8.p1.2.m2.1.1.cmml" xref="S8.p1.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p1.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S8.p1.2.m2.1d">italic_B</annotation></semantics></math>, are varied. A region of parameter space where there exists a finite faithful symbolic representation of purely hyperbolic dynamics is called a <span class="ltx_text ltx_font_italic" id="S8.p1.2.1">hyperbolic plateau</span>. In this study we will look at escape rates computed on two hyperbolic plateaus. In this section, we discuss how the edges of these plateaus are located.</p> </div> <div class="ltx_para" id="S8.p2"> <p class="ltx_p" id="S8.p2.6">As parameters are varied, homo/heteroclinic intersection points move continuously along the stable manifolds. Two, or more, intersection points can combine in a tangent bifurcation, i.e. when the stable and unstable manifolds tangentially intersect. This breaks hyperbolicity and produces global bifurcations in the dynamics. Keeping <math alttext="B" class="ltx_Math" display="inline" id="S8.p2.1.m1.1"><semantics id="S8.p2.1.m1.1a"><mi id="S8.p2.1.m1.1.1" xref="S8.p2.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S8.p2.1.m1.1b"><ci id="S8.p2.1.m1.1.1.cmml" xref="S8.p2.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S8.p2.1.m1.1d">italic_B</annotation></semantics></math> fixed at <math alttext="B=3.5" class="ltx_Math" display="inline" id="S8.p2.2.m2.1"><semantics id="S8.p2.2.m2.1a"><mrow id="S8.p2.2.m2.1.1" xref="S8.p2.2.m2.1.1.cmml"><mi id="S8.p2.2.m2.1.1.2" xref="S8.p2.2.m2.1.1.2.cmml">B</mi><mo id="S8.p2.2.m2.1.1.1" xref="S8.p2.2.m2.1.1.1.cmml">=</mo><mn id="S8.p2.2.m2.1.1.3" xref="S8.p2.2.m2.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S8.p2.2.m2.1b"><apply id="S8.p2.2.m2.1.1.cmml" xref="S8.p2.2.m2.1.1"><eq id="S8.p2.2.m2.1.1.1.cmml" xref="S8.p2.2.m2.1.1.1"></eq><ci id="S8.p2.2.m2.1.1.2.cmml" xref="S8.p2.2.m2.1.1.2">𝐵</ci><cn id="S8.p2.2.m2.1.1.3.cmml" type="float" xref="S8.p2.2.m2.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.2.m2.1c">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S8.p2.2.m2.1d">italic_B = 3.5</annotation></semantics></math> and varying <math alttext="E" class="ltx_Math" display="inline" id="S8.p2.3.m3.1"><semantics id="S8.p2.3.m3.1a"><mi id="S8.p2.3.m3.1.1" xref="S8.p2.3.m3.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S8.p2.3.m3.1b"><ci id="S8.p2.3.m3.1.1.cmml" xref="S8.p2.3.m3.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.3.m3.1c">E</annotation><annotation encoding="application/x-llamapun" id="S8.p2.3.m3.1d">italic_E</annotation></semantics></math> down from <math alttext="E=1" class="ltx_Math" display="inline" id="S8.p2.4.m4.1"><semantics id="S8.p2.4.m4.1a"><mrow id="S8.p2.4.m4.1.1" xref="S8.p2.4.m4.1.1.cmml"><mi id="S8.p2.4.m4.1.1.2" xref="S8.p2.4.m4.1.1.2.cmml">E</mi><mo id="S8.p2.4.m4.1.1.1" xref="S8.p2.4.m4.1.1.1.cmml">=</mo><mn id="S8.p2.4.m4.1.1.3" xref="S8.p2.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S8.p2.4.m4.1b"><apply id="S8.p2.4.m4.1.1.cmml" xref="S8.p2.4.m4.1.1"><eq id="S8.p2.4.m4.1.1.1.cmml" xref="S8.p2.4.m4.1.1.1"></eq><ci id="S8.p2.4.m4.1.1.2.cmml" xref="S8.p2.4.m4.1.1.2">𝐸</ci><cn id="S8.p2.4.m4.1.1.3.cmml" type="integer" xref="S8.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.4.m4.1c">E=1</annotation><annotation encoding="application/x-llamapun" id="S8.p2.4.m4.1d">italic_E = 1</annotation></semantics></math>, we observe a tangency at <math alttext="E=0.326\pm 0.006" class="ltx_Math" display="inline" id="S8.p2.5.m5.1"><semantics id="S8.p2.5.m5.1a"><mrow id="S8.p2.5.m5.1.1" xref="S8.p2.5.m5.1.1.cmml"><mi id="S8.p2.5.m5.1.1.2" xref="S8.p2.5.m5.1.1.2.cmml">E</mi><mo id="S8.p2.5.m5.1.1.1" xref="S8.p2.5.m5.1.1.1.cmml">=</mo><mrow id="S8.p2.5.m5.1.1.3" xref="S8.p2.5.m5.1.1.3.cmml"><mn id="S8.p2.5.m5.1.1.3.2" xref="S8.p2.5.m5.1.1.3.2.cmml">0.326</mn><mo id="S8.p2.5.m5.1.1.3.1" xref="S8.p2.5.m5.1.1.3.1.cmml">±</mo><mn id="S8.p2.5.m5.1.1.3.3" xref="S8.p2.5.m5.1.1.3.3.cmml">0.006</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p2.5.m5.1b"><apply id="S8.p2.5.m5.1.1.cmml" xref="S8.p2.5.m5.1.1"><eq id="S8.p2.5.m5.1.1.1.cmml" xref="S8.p2.5.m5.1.1.1"></eq><ci id="S8.p2.5.m5.1.1.2.cmml" xref="S8.p2.5.m5.1.1.2">𝐸</ci><apply id="S8.p2.5.m5.1.1.3.cmml" xref="S8.p2.5.m5.1.1.3"><csymbol cd="latexml" id="S8.p2.5.m5.1.1.3.1.cmml" xref="S8.p2.5.m5.1.1.3.1">plus-or-minus</csymbol><cn id="S8.p2.5.m5.1.1.3.2.cmml" type="float" xref="S8.p2.5.m5.1.1.3.2">0.326</cn><cn id="S8.p2.5.m5.1.1.3.3.cmml" type="float" xref="S8.p2.5.m5.1.1.3.3">0.006</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.5.m5.1c">E=0.326\pm 0.006</annotation><annotation encoding="application/x-llamapun" id="S8.p2.5.m5.1d">italic_E = 0.326 ± 0.006</annotation></semantics></math>. This marks the lower boundary of what we will refer to as the “upper” plateau. In this bifurcation, two intersection points of a “tip” of the unstable manifold merge and are eliminated as the tip retracts, as seen in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F6" title="Figure 6 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">6</span></a>e. Another tangency occurs at <math alttext="E=1.35\pm 0.05" class="ltx_Math" display="inline" id="S8.p2.6.m6.1"><semantics id="S8.p2.6.m6.1a"><mrow id="S8.p2.6.m6.1.1" xref="S8.p2.6.m6.1.1.cmml"><mi id="S8.p2.6.m6.1.1.2" xref="S8.p2.6.m6.1.1.2.cmml">E</mi><mo id="S8.p2.6.m6.1.1.1" xref="S8.p2.6.m6.1.1.1.cmml">=</mo><mrow id="S8.p2.6.m6.1.1.3" xref="S8.p2.6.m6.1.1.3.cmml"><mn id="S8.p2.6.m6.1.1.3.2" xref="S8.p2.6.m6.1.1.3.2.cmml">1.35</mn><mo id="S8.p2.6.m6.1.1.3.1" xref="S8.p2.6.m6.1.1.3.1.cmml">±</mo><mn id="S8.p2.6.m6.1.1.3.3" xref="S8.p2.6.m6.1.1.3.3.cmml">0.05</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p2.6.m6.1b"><apply id="S8.p2.6.m6.1.1.cmml" xref="S8.p2.6.m6.1.1"><eq id="S8.p2.6.m6.1.1.1.cmml" xref="S8.p2.6.m6.1.1.1"></eq><ci id="S8.p2.6.m6.1.1.2.cmml" xref="S8.p2.6.m6.1.1.2">𝐸</ci><apply id="S8.p2.6.m6.1.1.3.cmml" xref="S8.p2.6.m6.1.1.3"><csymbol cd="latexml" id="S8.p2.6.m6.1.1.3.1.cmml" xref="S8.p2.6.m6.1.1.3.1">plus-or-minus</csymbol><cn id="S8.p2.6.m6.1.1.3.2.cmml" type="float" xref="S8.p2.6.m6.1.1.3.2">1.35</cn><cn id="S8.p2.6.m6.1.1.3.3.cmml" type="float" xref="S8.p2.6.m6.1.1.3.3">0.05</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.6.m6.1c">E=1.35\pm 0.05</annotation><annotation encoding="application/x-llamapun" id="S8.p2.6.m6.1d">italic_E = 1.35 ± 0.05</annotation></semantics></math> that marks the top boundary of the upper plateau. In this bifurcation, one intersection point splits into three as a straight segment of the unstable manifold develops a cubic oscillation. This is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F6" title="Figure 6 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">6</span></a>g.</p> </div> <figure class="ltx_figure" id="S8.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="1022" id="S8.F6.g1" src="extracted/6294607/combinedTangenciesNEW.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S8.F6.6.3.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text" id="S8.F6.4.2" style="font-size:90%;">Tangent bifurcations as they occur near the edges of the two hyperbolic plateaus. <math alttext="B=3.5" class="ltx_Math" display="inline" id="S8.F6.3.1.m1.1"><semantics id="S8.F6.3.1.m1.1b"><mrow id="S8.F6.3.1.m1.1.1" xref="S8.F6.3.1.m1.1.1.cmml"><mi id="S8.F6.3.1.m1.1.1.2" xref="S8.F6.3.1.m1.1.1.2.cmml">B</mi><mo id="S8.F6.3.1.m1.1.1.1" xref="S8.F6.3.1.m1.1.1.1.cmml">=</mo><mn id="S8.F6.3.1.m1.1.1.3" xref="S8.F6.3.1.m1.1.1.3.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S8.F6.3.1.m1.1c"><apply id="S8.F6.3.1.m1.1.1.cmml" xref="S8.F6.3.1.m1.1.1"><eq id="S8.F6.3.1.m1.1.1.1.cmml" xref="S8.F6.3.1.m1.1.1.1"></eq><ci id="S8.F6.3.1.m1.1.1.2.cmml" xref="S8.F6.3.1.m1.1.1.2">𝐵</ci><cn id="S8.F6.3.1.m1.1.1.3.cmml" type="float" xref="S8.F6.3.1.m1.1.1.3">3.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.F6.3.1.m1.1d">B=3.5</annotation><annotation encoding="application/x-llamapun" id="S8.F6.3.1.m1.1e">italic_B = 3.5</annotation></semantics></math> and <math alttext="E" class="ltx_Math" display="inline" id="S8.F6.4.2.m2.1"><semantics id="S8.F6.4.2.m2.1b"><mi id="S8.F6.4.2.m2.1.1" xref="S8.F6.4.2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S8.F6.4.2.m2.1c"><ci id="S8.F6.4.2.m2.1.1.cmml" xref="S8.F6.4.2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.F6.4.2.m2.1d">E</annotation><annotation encoding="application/x-llamapun" id="S8.F6.4.2.m2.1e">italic_E</annotation></semantics></math> is allowed to vary. Stable manifolds are plotted in red, and unstable manifolds are plotted in blue. Manifolds with the same line style are computed at the same parameter value. (a) Tangency at the bottom of the ’lower’ plateau. (b) Resonance zone with zoomed in region from (a) outlined in black (c) Tangency at the top of the ’lower’ plateau. (d) Resonance zone with zoomed in region from (c) outlined in black (e) Tangency at the bottom of the ’upper’ plateau. (f) Resonance zone with zoomed in region from (e) outlined in black. (g) Cubic tangency at the top of the ’upper’ plateau. (h) Resonance zone with zoomed in region from (g) outlined in black</span></figcaption> </figure> <div class="ltx_para" id="S8.p3"> <p class="ltx_p" id="S8.p3.2">Ref. <cite class="ltx_cite ltx_citemacro_citep">Gonzalez and Jung, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib27" title="">2014</a></cite> presented a general topological analysis that connects the trellis in the upper hyperbolic plateau, i.e. a trellis having a full shift on three symbols, to a different trellis defining another hyperbolic plateau. This new trellis is no longer a full shift on three symbols but is a restricted finite shift on three symbols. It thus has a smaller topological entropy. The general topological nature of this prior work implies that it should be possible to vary our parameters <math alttext="E" class="ltx_Math" display="inline" id="S8.p3.1.m1.1"><semantics id="S8.p3.1.m1.1a"><mi id="S8.p3.1.m1.1.1" xref="S8.p3.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S8.p3.1.m1.1b"><ci id="S8.p3.1.m1.1.1.cmml" xref="S8.p3.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p3.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S8.p3.1.m1.1d">italic_E</annotation></semantics></math> and/or <math alttext="B" class="ltx_Math" display="inline" id="S8.p3.2.m2.1"><semantics id="S8.p3.2.m2.1a"><mi id="S8.p3.2.m2.1.1" xref="S8.p3.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S8.p3.2.m2.1b"><ci id="S8.p3.2.m2.1.1.cmml" xref="S8.p3.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p3.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S8.p3.2.m2.1d">italic_B</annotation></semantics></math> in such a way as to identify this new trellis in the atomic system.</p> </div> <div class="ltx_para" id="S8.p4"> <p class="ltx_p" id="S8.p4.4">We hunted for this trellis by numerically computing and plotting the stable and unstable manifolds at different parameter values. Based on the trellis topology described in Ref. <cite class="ltx_cite ltx_citemacro_citep">Gonzalez and Jung, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib27" title="">2014</a></cite>, we knew what tangent bifurcations would lead to the new trellis. Then, a binary search of parameter space was conducted using manifold tangencies as heuristics. Continuing to vary <math alttext="E" class="ltx_Math" display="inline" id="S8.p4.1.m1.1"><semantics id="S8.p4.1.m1.1a"><mi id="S8.p4.1.m1.1.1" xref="S8.p4.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S8.p4.1.m1.1b"><ci id="S8.p4.1.m1.1.1.cmml" xref="S8.p4.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p4.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S8.p4.1.m1.1d">italic_E</annotation></semantics></math> downward, the first tangency occurs at <math alttext="E=0.295\pm 0.01" class="ltx_Math" display="inline" id="S8.p4.2.m2.1"><semantics id="S8.p4.2.m2.1a"><mrow id="S8.p4.2.m2.1.1" xref="S8.p4.2.m2.1.1.cmml"><mi id="S8.p4.2.m2.1.1.2" xref="S8.p4.2.m2.1.1.2.cmml">E</mi><mo id="S8.p4.2.m2.1.1.1" xref="S8.p4.2.m2.1.1.1.cmml">=</mo><mrow id="S8.p4.2.m2.1.1.3" xref="S8.p4.2.m2.1.1.3.cmml"><mn id="S8.p4.2.m2.1.1.3.2" xref="S8.p4.2.m2.1.1.3.2.cmml">0.295</mn><mo id="S8.p4.2.m2.1.1.3.1" xref="S8.p4.2.m2.1.1.3.1.cmml">±</mo><mn id="S8.p4.2.m2.1.1.3.3" xref="S8.p4.2.m2.1.1.3.3.cmml">0.01</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p4.2.m2.1b"><apply id="S8.p4.2.m2.1.1.cmml" xref="S8.p4.2.m2.1.1"><eq id="S8.p4.2.m2.1.1.1.cmml" xref="S8.p4.2.m2.1.1.1"></eq><ci id="S8.p4.2.m2.1.1.2.cmml" xref="S8.p4.2.m2.1.1.2">𝐸</ci><apply id="S8.p4.2.m2.1.1.3.cmml" xref="S8.p4.2.m2.1.1.3"><csymbol cd="latexml" id="S8.p4.2.m2.1.1.3.1.cmml" xref="S8.p4.2.m2.1.1.3.1">plus-or-minus</csymbol><cn id="S8.p4.2.m2.1.1.3.2.cmml" type="float" xref="S8.p4.2.m2.1.1.3.2">0.295</cn><cn id="S8.p4.2.m2.1.1.3.3.cmml" type="float" xref="S8.p4.2.m2.1.1.3.3">0.01</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p4.2.m2.1c">E=0.295\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S8.p4.2.m2.1d">italic_E = 0.295 ± 0.01</annotation></semantics></math> marking the top border of the “lower” plateau, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F6" title="Figure 6 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">6</span></a>c. Further varying <math alttext="E" class="ltx_Math" display="inline" id="S8.p4.3.m3.1"><semantics id="S8.p4.3.m3.1a"><mi id="S8.p4.3.m3.1.1" xref="S8.p4.3.m3.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S8.p4.3.m3.1b"><ci id="S8.p4.3.m3.1.1.cmml" xref="S8.p4.3.m3.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p4.3.m3.1c">E</annotation><annotation encoding="application/x-llamapun" id="S8.p4.3.m3.1d">italic_E</annotation></semantics></math>, the tangency that marks the lower boundary occurs at <math alttext="E=0.275\pm 0.003" class="ltx_Math" display="inline" id="S8.p4.4.m4.1"><semantics id="S8.p4.4.m4.1a"><mrow id="S8.p4.4.m4.1.1" xref="S8.p4.4.m4.1.1.cmml"><mi id="S8.p4.4.m4.1.1.2" xref="S8.p4.4.m4.1.1.2.cmml">E</mi><mo id="S8.p4.4.m4.1.1.1" xref="S8.p4.4.m4.1.1.1.cmml">=</mo><mrow id="S8.p4.4.m4.1.1.3" xref="S8.p4.4.m4.1.1.3.cmml"><mn id="S8.p4.4.m4.1.1.3.2" xref="S8.p4.4.m4.1.1.3.2.cmml">0.275</mn><mo id="S8.p4.4.m4.1.1.3.1" xref="S8.p4.4.m4.1.1.3.1.cmml">±</mo><mn id="S8.p4.4.m4.1.1.3.3" xref="S8.p4.4.m4.1.1.3.3.cmml">0.003</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p4.4.m4.1b"><apply id="S8.p4.4.m4.1.1.cmml" xref="S8.p4.4.m4.1.1"><eq id="S8.p4.4.m4.1.1.1.cmml" xref="S8.p4.4.m4.1.1.1"></eq><ci id="S8.p4.4.m4.1.1.2.cmml" xref="S8.p4.4.m4.1.1.2">𝐸</ci><apply id="S8.p4.4.m4.1.1.3.cmml" xref="S8.p4.4.m4.1.1.3"><csymbol cd="latexml" id="S8.p4.4.m4.1.1.3.1.cmml" xref="S8.p4.4.m4.1.1.3.1">plus-or-minus</csymbol><cn id="S8.p4.4.m4.1.1.3.2.cmml" type="float" xref="S8.p4.4.m4.1.1.3.2">0.275</cn><cn id="S8.p4.4.m4.1.1.3.3.cmml" type="float" xref="S8.p4.4.m4.1.1.3.3">0.003</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p4.4.m4.1c">E=0.275\pm 0.003</annotation><annotation encoding="application/x-llamapun" id="S8.p4.4.m4.1d">italic_E = 0.275 ± 0.003</annotation></semantics></math> (Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F6" title="Figure 6 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">6</span></a>a). Both of these bifurcations involve a tip retracting and two intersection points merging and disappearing.</p> </div> <div class="ltx_para" id="S8.p5"> <p class="ltx_p" id="S8.p5.1">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F7" title="Figure 7 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">7</span></a>a shows the trellis in the lower plateau, with Markov partition rectangles labelled ‘0’ through ‘10’. We identified these partition rectangles using the homotopic lobe dynamics approach in Ref. <cite class="ltx_cite ltx_citemacro_citep">Mitchell, <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#bib.bib20" title="">2012</a></cite>. Iterating each of these rectangles forward and observing their intersections with the original rectangles, one can construct the transition graph between the Markov rectangles (as in Figs. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>a and <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.F3" title="Figure 3 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">3</span></a>b). Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F7" title="Figure 7 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">7</span></a>b shows this new transition graph. One can also use the homotopic lobe dynamics approach to directly obtain this transition graph.</p> </div> <figure class="ltx_figure" id="S8.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="224" id="S8.F7.g1" src="extracted/6294607/combinedLowerTrellis.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S8.F7.4.2.1" style="font-size:90%;">Figure 7</span>: </span><span class="ltx_text" id="S8.F7.2.1" style="font-size:90%;">Symbolic dyanmics of the lower hyperbolic plateau. (a) Trellis at <math alttext="E=0.285" class="ltx_Math" display="inline" id="S8.F7.2.1.m1.1"><semantics id="S8.F7.2.1.m1.1b"><mrow id="S8.F7.2.1.m1.1.1" xref="S8.F7.2.1.m1.1.1.cmml"><mi id="S8.F7.2.1.m1.1.1.2" xref="S8.F7.2.1.m1.1.1.2.cmml">E</mi><mo id="S8.F7.2.1.m1.1.1.1" xref="S8.F7.2.1.m1.1.1.1.cmml">=</mo><mn id="S8.F7.2.1.m1.1.1.3" xref="S8.F7.2.1.m1.1.1.3.cmml">0.285</mn></mrow><annotation-xml encoding="MathML-Content" id="S8.F7.2.1.m1.1c"><apply id="S8.F7.2.1.m1.1.1.cmml" xref="S8.F7.2.1.m1.1.1"><eq id="S8.F7.2.1.m1.1.1.1.cmml" xref="S8.F7.2.1.m1.1.1.1"></eq><ci id="S8.F7.2.1.m1.1.1.2.cmml" xref="S8.F7.2.1.m1.1.1.2">𝐸</ci><cn id="S8.F7.2.1.m1.1.1.3.cmml" type="float" xref="S8.F7.2.1.m1.1.1.3">0.285</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.F7.2.1.m1.1d">E=0.285</annotation><annotation encoding="application/x-llamapun" id="S8.F7.2.1.m1.1e">italic_E = 0.285</annotation></semantics></math> with partition rectangles colored and labelled. Two forward iterates of the unstable manifold are plotted (blue) and two backward iterates of the stable manifold are plotted (red). (b) Transition graph representing the symbolic dynamics extracted from (a) using homotopic lobe dynamics.</span></figcaption> </figure> </section> <section class="ltx_section" id="S9"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IX </span>Varying the electron energy</h2> <div class="ltx_para" id="S9.p1"> <p class="ltx_p" id="S9.p1.1">Periodic orbits change continuously as parameters are varied, so long as they do not disappear in a bifurcation. This allows us to easily numerically continue them through parameter space. A continuation is done by making small perturbations of the parameter values and using the previously computed orbits as initial guesses for a Newton’s method solver. In these cases Newton’s method converges quickly because periodic orbits change continuously.</p> </div> <div class="ltx_para" id="S9.p2"> <p class="ltx_p" id="S9.p2.4">We first continue orbits starting from <math alttext="E=1" class="ltx_Math" display="inline" id="S9.p2.1.m1.1"><semantics id="S9.p2.1.m1.1a"><mrow id="S9.p2.1.m1.1.1" xref="S9.p2.1.m1.1.1.cmml"><mi id="S9.p2.1.m1.1.1.2" xref="S9.p2.1.m1.1.1.2.cmml">E</mi><mo id="S9.p2.1.m1.1.1.1" xref="S9.p2.1.m1.1.1.1.cmml">=</mo><mn id="S9.p2.1.m1.1.1.3" xref="S9.p2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S9.p2.1.m1.1b"><apply id="S9.p2.1.m1.1.1.cmml" xref="S9.p2.1.m1.1.1"><eq id="S9.p2.1.m1.1.1.1.cmml" xref="S9.p2.1.m1.1.1.1"></eq><ci id="S9.p2.1.m1.1.1.2.cmml" xref="S9.p2.1.m1.1.1.2">𝐸</ci><cn id="S9.p2.1.m1.1.1.3.cmml" type="integer" xref="S9.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p2.1.m1.1c">E=1</annotation><annotation encoding="application/x-llamapun" id="S9.p2.1.m1.1d">italic_E = 1</annotation></semantics></math> on the upper hyperbolic plateau up towards the top of the plateau near the cubic tangent bifurcation we observed. Next, we continue the orbits to the bottom of the upper plateau at <math alttext="E=0.326" class="ltx_Math" display="inline" id="S9.p2.2.m2.1"><semantics id="S9.p2.2.m2.1a"><mrow id="S9.p2.2.m2.1.1" xref="S9.p2.2.m2.1.1.cmml"><mi id="S9.p2.2.m2.1.1.2" xref="S9.p2.2.m2.1.1.2.cmml">E</mi><mo id="S9.p2.2.m2.1.1.1" xref="S9.p2.2.m2.1.1.1.cmml">=</mo><mn id="S9.p2.2.m2.1.1.3" xref="S9.p2.2.m2.1.1.3.cmml">0.326</mn></mrow><annotation-xml encoding="MathML-Content" id="S9.p2.2.m2.1b"><apply id="S9.p2.2.m2.1.1.cmml" xref="S9.p2.2.m2.1.1"><eq id="S9.p2.2.m2.1.1.1.cmml" xref="S9.p2.2.m2.1.1.1"></eq><ci id="S9.p2.2.m2.1.1.2.cmml" xref="S9.p2.2.m2.1.1.2">𝐸</ci><cn id="S9.p2.2.m2.1.1.3.cmml" type="float" xref="S9.p2.2.m2.1.1.3">0.326</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p2.2.m2.1c">E=0.326</annotation><annotation encoding="application/x-llamapun" id="S9.p2.2.m2.1d">italic_E = 0.326</annotation></semantics></math>. At this point we have periodic orbits computed across the range of the upper plateau. Next, we continue the orbits from the upper plateau down to the top of the lower hyperbolic plateau located at <math alttext="E=0.295" class="ltx_Math" display="inline" id="S9.p2.3.m3.1"><semantics id="S9.p2.3.m3.1a"><mrow id="S9.p2.3.m3.1.1" xref="S9.p2.3.m3.1.1.cmml"><mi id="S9.p2.3.m3.1.1.2" xref="S9.p2.3.m3.1.1.2.cmml">E</mi><mo id="S9.p2.3.m3.1.1.1" xref="S9.p2.3.m3.1.1.1.cmml">=</mo><mn id="S9.p2.3.m3.1.1.3" xref="S9.p2.3.m3.1.1.3.cmml">0.295</mn></mrow><annotation-xml encoding="MathML-Content" id="S9.p2.3.m3.1b"><apply id="S9.p2.3.m3.1.1.cmml" xref="S9.p2.3.m3.1.1"><eq id="S9.p2.3.m3.1.1.1.cmml" xref="S9.p2.3.m3.1.1.1"></eq><ci id="S9.p2.3.m3.1.1.2.cmml" xref="S9.p2.3.m3.1.1.2">𝐸</ci><cn id="S9.p2.3.m3.1.1.3.cmml" type="float" xref="S9.p2.3.m3.1.1.3">0.295</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p2.3.m3.1c">E=0.295</annotation><annotation encoding="application/x-llamapun" id="S9.p2.3.m3.1d">italic_E = 0.295</annotation></semantics></math>. A set of bifurcations occur between the plateaus that we discuss below. Finally, we continue the orbits through the lower plateau down to <math alttext="E=0.275" class="ltx_Math" display="inline" id="S9.p2.4.m4.1"><semantics id="S9.p2.4.m4.1a"><mrow id="S9.p2.4.m4.1.1" xref="S9.p2.4.m4.1.1.cmml"><mi id="S9.p2.4.m4.1.1.2" xref="S9.p2.4.m4.1.1.2.cmml">E</mi><mo id="S9.p2.4.m4.1.1.1" xref="S9.p2.4.m4.1.1.1.cmml">=</mo><mn id="S9.p2.4.m4.1.1.3" xref="S9.p2.4.m4.1.1.3.cmml">0.275</mn></mrow><annotation-xml encoding="MathML-Content" id="S9.p2.4.m4.1b"><apply id="S9.p2.4.m4.1.1.cmml" xref="S9.p2.4.m4.1.1"><eq id="S9.p2.4.m4.1.1.1.cmml" xref="S9.p2.4.m4.1.1.1"></eq><ci id="S9.p2.4.m4.1.1.2.cmml" xref="S9.p2.4.m4.1.1.2">𝐸</ci><cn id="S9.p2.4.m4.1.1.3.cmml" type="float" xref="S9.p2.4.m4.1.1.3">0.275</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p2.4.m4.1c">E=0.275</annotation><annotation encoding="application/x-llamapun" id="S9.p2.4.m4.1d">italic_E = 0.275</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S9.p3"> <p class="ltx_p" id="S9.p3.1">When we numerically continue orbits from the upper plateau to the lower plateau, we observe bifurcations in the set of periodic orbits. These bifurcations indicate global changes in the dynamics that correspond to topological changes in the trellis due to tangent bifurcations. Some orbits from the upper plateau disappear before reaching the lower plateau. However, no new orbits are created in the lower plateau. That is, all periodic orbits in the lower plateau are a subset of the orbits in the upper plateau.</p> </div> <div class="ltx_para" id="S9.p4"> <p class="ltx_p" id="S9.p4.1">To determine which orbits to throw out and which orbits to keep, we formally continue them all to the lower plateau and then check that each of these orbits is truly a periodic orbit. We do this by mapping an orbit forward and checking that it remains the same. Periodic orbits that have disappeared in a bifurcation may still produce an orbit that is close to periodic, but deviates exponentially upon iteration. Orbits that deviate in this way are removed from the set of periodic orbits. From the symbolic dynamics of the lower plateau, the number of periodic orbits for a given itinerary length is known. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S6.T1" title="Table 1 ‣ VI Computing Periodic Orbits via Phase Space Partitioning ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">1</span></a> shows the number of orbits on the lower plateau compared to the upper one. After the continuation and orbit pruning we find a set of orbits with the correct distribution of discrete periods predicted from the symbolic dynamics (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S8.F7" title="Figure 7 ‣ VIII Locating Hyperbolic Plateaus ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">7</span></a>b).</p> </div> <div class="ltx_para" id="S9.p5"> <p class="ltx_p" id="S9.p5.2">The discrete spectral determinant computations are shown over Monte Carlo simulations in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S9.F8" title="Figure 8 ‣ IX Varying the electron energy ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">8</span></a>a across both hyperbolic plateaus. The confidence intervals are computed as the difference between using all orbits up to <math alttext="n_{p}=10" class="ltx_Math" display="inline" id="S9.p5.1.m1.1"><semantics id="S9.p5.1.m1.1a"><mrow id="S9.p5.1.m1.1.1" xref="S9.p5.1.m1.1.1.cmml"><msub id="S9.p5.1.m1.1.1.2" xref="S9.p5.1.m1.1.1.2.cmml"><mi id="S9.p5.1.m1.1.1.2.2" xref="S9.p5.1.m1.1.1.2.2.cmml">n</mi><mi id="S9.p5.1.m1.1.1.2.3" xref="S9.p5.1.m1.1.1.2.3.cmml">p</mi></msub><mo id="S9.p5.1.m1.1.1.1" xref="S9.p5.1.m1.1.1.1.cmml">=</mo><mn id="S9.p5.1.m1.1.1.3" xref="S9.p5.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S9.p5.1.m1.1b"><apply id="S9.p5.1.m1.1.1.cmml" xref="S9.p5.1.m1.1.1"><eq id="S9.p5.1.m1.1.1.1.cmml" xref="S9.p5.1.m1.1.1.1"></eq><apply id="S9.p5.1.m1.1.1.2.cmml" xref="S9.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S9.p5.1.m1.1.1.2.1.cmml" xref="S9.p5.1.m1.1.1.2">subscript</csymbol><ci id="S9.p5.1.m1.1.1.2.2.cmml" xref="S9.p5.1.m1.1.1.2.2">𝑛</ci><ci id="S9.p5.1.m1.1.1.2.3.cmml" xref="S9.p5.1.m1.1.1.2.3">𝑝</ci></apply><cn id="S9.p5.1.m1.1.1.3.cmml" type="integer" xref="S9.p5.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p5.1.m1.1c">n_{p}=10</annotation><annotation encoding="application/x-llamapun" id="S9.p5.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 10</annotation></semantics></math> and <math alttext="n_{p}=9" class="ltx_Math" display="inline" id="S9.p5.2.m2.1"><semantics id="S9.p5.2.m2.1a"><mrow id="S9.p5.2.m2.1.1" xref="S9.p5.2.m2.1.1.cmml"><msub id="S9.p5.2.m2.1.1.2" xref="S9.p5.2.m2.1.1.2.cmml"><mi id="S9.p5.2.m2.1.1.2.2" xref="S9.p5.2.m2.1.1.2.2.cmml">n</mi><mi id="S9.p5.2.m2.1.1.2.3" xref="S9.p5.2.m2.1.1.2.3.cmml">p</mi></msub><mo id="S9.p5.2.m2.1.1.1" xref="S9.p5.2.m2.1.1.1.cmml">=</mo><mn id="S9.p5.2.m2.1.1.3" xref="S9.p5.2.m2.1.1.3.cmml">9</mn></mrow><annotation-xml encoding="MathML-Content" id="S9.p5.2.m2.1b"><apply id="S9.p5.2.m2.1.1.cmml" xref="S9.p5.2.m2.1.1"><eq id="S9.p5.2.m2.1.1.1.cmml" xref="S9.p5.2.m2.1.1.1"></eq><apply id="S9.p5.2.m2.1.1.2.cmml" xref="S9.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S9.p5.2.m2.1.1.2.1.cmml" xref="S9.p5.2.m2.1.1.2">subscript</csymbol><ci id="S9.p5.2.m2.1.1.2.2.cmml" xref="S9.p5.2.m2.1.1.2.2">𝑛</ci><ci id="S9.p5.2.m2.1.1.2.3.cmml" xref="S9.p5.2.m2.1.1.2.3">𝑝</ci></apply><cn id="S9.p5.2.m2.1.1.3.cmml" type="integer" xref="S9.p5.2.m2.1.1.3">9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p5.2.m2.1c">n_{p}=9</annotation><annotation encoding="application/x-llamapun" id="S9.p5.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 9</annotation></semantics></math>. The convergence of the discrete spectral determinant at three parameter values is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>a, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>c, and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>e. All parameter values show accurate convergence and our methods appear to be capturing the totality of the discrete dynamics. Notice the difference between the computations at the tops of each hyperbolic plateau. At the top of the upper plateau the escape rate drops precipitously towards zero. We do not see the same behavior at the top of the lower plateau. We suspect this is due to the cubic nature of the tangency on the upper plateau, but we have not precisely determined the true cause.</p> </div> <div class="ltx_para" id="S9.p6"> <p class="ltx_p" id="S9.p6.1">The continuous spectral determinant computations are shown over Monte Carlo simulations in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S9.F8" title="Figure 8 ‣ IX Varying the electron energy ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">8</span></a>b across both hyperbolic plateaus. The confidence intervals are computed in the same way as for the discrete case. The convergence of the continuous spectral determinant at three parameter values is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>b, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>d, and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.15710v1#S7.F5" title="Figure 5 ‣ VII Escape Rate from Periodic Orbits ‣ Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen"><span class="ltx_text ltx_ref_tag">5</span></a>f. There is excellent convergence on the lower plateau and the bottom of the upper plateau. However, compared to the discrete case, the continuous spectral determinant at the top of the upper plateau begins to converge more slowly. This implies a larger set of periodic orbits is needed in that region to fully capture the escape dynamics. We suspect this is due to the cubic nature of the tangency at the top of the plateau. Likely long orbits that shadow heteroclinic orbits become more important as you get closer to a cubic tangency, but this is still a conjecture.</p> </div> <figure class="ltx_figure" id="S9.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="819" id="S9.F8.g1" src="extracted/6294607/combined.png" width="479"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S9.F8.2.1.1" style="font-size:90%;">Figure 8</span>: </span><span class="ltx_text" id="S9.F8.3.2" style="font-size:90%;">(a) Discrete time escape rate computed with Monte Carlo methods in black and periodic orbit methods in red. (b) Continuous time escape rate computed with Monte Carlo methods in blue and periodic orbit methods in magenta. Black vertical line marks the bottom boundary of the upper plateau. Blue vertical lines mark the boundaries of the lower plateau with error bounds illustrated by blue rectangles.</span></figcaption> </figure> </section> <section class="ltx_section" id="S10"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">X </span>Conclusion</h2> <div class="ltx_para" id="S10.p1"> <p class="ltx_p" id="S10.p1.1">Periodic orbits act as the skeleton of a dynamical system, and spectral determinants can be used to compute escape rates from periodic orbits. Escape rates computed in this way may use orders of magnitude fewer trajectories than Monte Carlo simulations, in this study four orders of magnitude fewer. A second advantage to periodic orbit methods is that they do not have to be entirely recomputed for a change in parameters. In a chaotic system even a small change in parameters requires an entirely new Monte Carlo simulation to accurately compute escape. On the other hand, periodic orbits computed for one parameter can be numerically continued to another parameter value rather than recomputing all the orbits from scratch.</p> </div> <div class="ltx_para" id="S10.p2"> <p class="ltx_p" id="S10.p2.1">Additionally, the periodic orbit methods presented in this paper allow for a thorough analysis of the system and provides insights that perhaps would not have been noticed otherwise. Periodic orbit theory provides a robust connection between symbolic dynamics and periodic orbits, which in conjunction with phase space partitions allows for the detailed probing of dynamics. Analyzing heteroclinic tangles tells us about global bifurcations, and bifurcations in periodic orbits indicate changes in phase space topology. Further analysis of symmetry and its role in bifurcations, though not mentioned in detail here, can provide yet another layer of understanding to chaotic systems. Furthermore, these methods also have applications to semi-classical analyses of open quantum systems. By including quantum properties of periodic orbits in the spectral determinant one can relate the zeros of the spectral determinant directly to quantum resonances. The breadth of systems that periodic orbit techniques can be applied to is still not known, but this paper expands the applications to additional experimentally testable systems.</p> </div> <div class="ltx_acknowledgements"> <h6 class="ltx_title ltx_title_acknowledgements">Acknowledgements.</h6> The computation of the Monte Carlo simulations, Lyopanov exponents, continuous time periods, and periodic orbit continuations was done on the PINNACLES cluster: Intel-28-Core Xeon Gold 6330 2.0GHz nodes </div> </section> <section class="ltx_section" id="S11"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">XI </span>Author Declarations</h2> <section class="ltx_subsection" id="S11.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">XI.1 </span>Conflict of Interest</h3> <div class="ltx_para" id="S11.SS1.p1"> <p class="ltx_p" id="S11.SS1.p1.1">The authors have no conflicts to disclose</p> </div> </section> <section class="ltx_subsection" id="S11.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">XI.2 </span>Author Contributions</h3> <div class="ltx_para" id="S11.SS2.p1"> <p class="ltx_p" id="S11.SS2.p1.1"><span class="ltx_text ltx_font_bold" id="S11.SS2.p1.1.1">Ethan Custodio:</span> Conceptualization (equal), Data Curation (lead), Formal Analysis (lead), Methodology (equal), Resources (lead), Software (lead), Validation (equal), Visualization (lead), Writing/Original Draft Preparation (lead), Writing/Review & Editing (equal)</p> </div> <div class="ltx_para" id="S11.SS2.p2"> <p class="ltx_p" id="S11.SS2.p2.1"><span class="ltx_text ltx_font_bold" id="S11.SS2.p2.1.1">Sulimon Sattari:</span> Conceptualization (equal), Formal Analysis (support), Methodology (equal), Writing/Review & Editing (equal)</p> </div> <div class="ltx_para" id="S11.SS2.p3"> <p class="ltx_p" id="S11.SS2.p3.1"><span class="ltx_text ltx_font_bold" id="S11.SS2.p3.1.1">Kevin Mitchell:</span> Conceptualization (equal), Formal Analysis (support), Methodology (equal), Resources (support), Supervision (lead), Validation (equal), Writing/Review & Editing (equal)</p> </div> </section> </section> <section class="ltx_section" id="S12"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">XII </span>Data Availability</h2> <div class="ltx_para" id="S12.p1"> <p class="ltx_p" id="S12.p1.1">The data that support the findings of this study are available from the corresponding author upon reasonable request.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Poincaré (1890)</span> <span class="ltx_bibblock">H. 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