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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.15937">arXiv:2412.15937</a> <span> [<a href="https://arxiv.org/pdf/2412.15937">pdf</a>, <a href="https://arxiv.org/ps/2412.15937">ps</a>, <a href="https://arxiv.org/format/2412.15937">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Spectral comparison results for Laplacians on discrete graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bifulco%2C+P">Patrizio Bifulco</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Rose%2C+C">Christian Rose</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.15937v2-abstract-short" style="display: inline;"> In the recent literature, various authors have studied spectral comparison results for Schr枚dinger operators with discrete spectrum in different settings including Euclidean domains and quantum graphs. In this note we derive such spectral comparison results in a rather general framework for general and possibly infinite discrete graphs. Along the way, we establish a discrete version of the local W… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.15937v2-abstract-full').style.display = 'inline'; document.getElementById('2412.15937v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.15937v2-abstract-full" style="display: none;"> In the recent literature, various authors have studied spectral comparison results for Schr枚dinger operators with discrete spectrum in different settings including Euclidean domains and quantum graphs. In this note we derive such spectral comparison results in a rather general framework for general and possibly infinite discrete graphs. Along the way, we establish a discrete version of the local Weyl law whose proof does neither involve any Tauberian theorem nor the Weyl law as used in the continuous case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.15937v2-abstract-full').style.display = 'none'; document.getElementById('2412.15937v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47A75; 47B93; 81Q35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14965">arXiv:2411.14965</a> <span> [<a href="https://arxiv.org/pdf/2411.14965">pdf</a>, <a href="https://arxiv.org/ps/2411.14965">ps</a>, <a href="https://arxiv.org/format/2411.14965">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> The curious spectra and dynamics of non-locally finite crystals </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Post%2C+O">Olaf Post</a>, <a href="/search/math-ph?searchtype=author&query=Sabri%2C+M">Mostafa Sabri</a>, <a href="/search/math-ph?searchtype=author&query=T%C3%A4ufer%2C+M">Matthias T盲ufer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14965v1-abstract-short" style="display: inline;"> This paper is devoted to the investigation of the spectral theory and dynamical properties of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather intriguing behaviour: for example, we construct a periodic graph whose Laplacian has purely singular continuous spectrum. Regarding point spectrum, and differe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14965v1-abstract-full').style.display = 'inline'; document.getElementById('2411.14965v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14965v1-abstract-full" style="display: none;"> This paper is devoted to the investigation of the spectral theory and dynamical properties of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather intriguing behaviour: for example, we construct a periodic graph whose Laplacian has purely singular continuous spectrum. Regarding point spectrum, and different to the locally finite case, we construct a graph with a partly flat band whose eigenvectors must have infinite support. Concerning dynamical aspects, under some assumptions we prove that motion remains ballistic along at least one layer. We also construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This provides a negative answer to an open question in this context. Furthermore, we include a discussion of the fractional Laplacian for which we prove a phase transition in its dynamical behaviour. Generally speaking, many questions still remain open, and we believe that the studied class of graphs can serve as a playground to better understand exotic spectra and dynamics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14965v1-abstract-full').style.display = 'none'; document.getElementById('2411.14965v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C50; 39A12; 58J50; 81Q10; 81Q35; 81Q80 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.21719">arXiv:2407.21719</a> <span> [<a href="https://arxiv.org/pdf/2407.21719">pdf</a>, <a href="https://arxiv.org/ps/2407.21719">ps</a>, <a href="https://arxiv.org/format/2407.21719">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> A modified local Weyl law and spectral comparison results for $未'$-coupling conditions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bifulco%2C+P">Patrizio Bifulco</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.21719v1-abstract-short" style="display: inline;"> We study Schr枚dinger operators on compact finite metric graphs subject to $未'$-coupling conditions. Based on a novel modified local Weyl law, we derive an explicit expression for the limiting mean eigenvalue distance of two different self-adjoint realisations on a given graph. Furthermore, using this spectral comparison result, we also study the limiting mean eigenvalue distance comparing $未'$-cou… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.21719v1-abstract-full').style.display = 'inline'; document.getElementById('2407.21719v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.21719v1-abstract-full" style="display: none;"> We study Schr枚dinger operators on compact finite metric graphs subject to $未'$-coupling conditions. Based on a novel modified local Weyl law, we derive an explicit expression for the limiting mean eigenvalue distance of two different self-adjoint realisations on a given graph. Furthermore, using this spectral comparison result, we also study the limiting mean eigenvalue distance comparing $未'$-coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. . <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.21719v1-abstract-full').style.display = 'none'; document.getElementById('2407.21719v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, 1 figure; comments are welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L05; 81Q35; 34L15; 34L20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.14357">arXiv:2312.14357</a> <span> [<a href="https://arxiv.org/pdf/2312.14357">pdf</a>, <a href="https://arxiv.org/ps/2312.14357">ps</a>, <a href="https://arxiv.org/format/2312.14357">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.matpur.2024.06.009">10.1016/j.matpur.2024.06.009 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On Bose-Einstein condensation in interacting Bose gases in the Kac-Luttinger model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Boccato%2C+C">Chiara Boccato</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pechmann%2C+M">Maximilian Pechmann</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.14357v1-abstract-short" style="display: inline;"> We study interacting Bose gases of dimensions $2\le d \in \mathbb N$ at zero temperature in a random model known as the Kac-Luttinger model. Choosing the pair-interaction between the bosons to be of a mean-field type, we prove (complete) Bose-Einstein condensation in probability or with probability almost one into the minimizer of a Hartree-type functional. We accomplish this by building upon very… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.14357v1-abstract-full').style.display = 'inline'; document.getElementById('2312.14357v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.14357v1-abstract-full" style="display: none;"> We study interacting Bose gases of dimensions $2\le d \in \mathbb N$ at zero temperature in a random model known as the Kac-Luttinger model. Choosing the pair-interaction between the bosons to be of a mean-field type, we prove (complete) Bose-Einstein condensation in probability or with probability almost one into the minimizer of a Hartree-type functional. We accomplish this by building upon very recent results by Alain-Sol Sznitman on the spectral gap of the noninteracting Bose gas. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.14357v1-abstract-full').style.display = 'none'; document.getElementById('2312.14357v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B44; 60G55; 81V70; 82B10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Pures Appl. (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.16869">arXiv:2308.16869</a> <span> [<a href="https://arxiv.org/pdf/2308.16869">pdf</a>, <a href="https://arxiv.org/ps/2308.16869">ps</a>, <a href="https://arxiv.org/format/2308.16869">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/5.0178226">10.1063/5.0178226 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Some spectral comparison results on infinite quantum graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bifulco%2C+P">Patrizio Bifulco</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.16869v1-abstract-short" style="display: inline;"> In this paper we establish spectral comparison results for Schr枚dinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite quantum graphs such as a modified local Weyl law. In this sense, we regard this paper as a starting point for a more thorough investigation of spectral comparison res… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.16869v1-abstract-full').style.display = 'inline'; document.getElementById('2308.16869v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.16869v1-abstract-full" style="display: none;"> In this paper we establish spectral comparison results for Schr枚dinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite quantum graphs such as a modified local Weyl law. In this sense, we regard this paper as a starting point for a more thorough investigation of spectral comparison results on more general infinite metric graphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.16869v1-abstract-full').style.display = 'none'; document.getElementById('2308.16869v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages, 3 figures, comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L05; 81Q35; 34L15; 34L20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Phys. (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.10881">arXiv:2308.10881</a> <span> [<a href="https://arxiv.org/pdf/2308.10881">pdf</a>, <a href="https://arxiv.org/ps/2308.10881">ps</a>, <a href="https://arxiv.org/format/2308.10881">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00013-024-01997-9">10.1007/s00013-024-01997-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A note on Ambarzumian's theorem for quantum graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bifulco%2C+P">Patrizio Bifulco</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.10881v1-abstract-short" style="display: inline;"> Based on the main result presented in a recent paper, we derive Ambarzumian-type theorems for Schr枚dinger operators defined on quantum graphs. We recover existing results such as the classical theorem by Ambarzumian and establish some seemingly new statements, too. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.10881v1-abstract-full" style="display: none;"> Based on the main result presented in a recent paper, we derive Ambarzumian-type theorems for Schr枚dinger operators defined on quantum graphs. We recover existing results such as the classical theorem by Ambarzumian and establish some seemingly new statements, too. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.10881v1-abstract-full').style.display = 'none'; document.getElementById('2308.10881v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 pages, comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L05; 81Q35; 34L15; 34L20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Archiv der Mathematik (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2301.05076">arXiv:2301.05076</a> <span> [<a href="https://arxiv.org/pdf/2301.05076">pdf</a>, <a href="https://arxiv.org/ps/2301.05076">ps</a>, <a href="https://arxiv.org/format/2301.05076">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00023-023-01399-7">10.1007/s00023-023-01399-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=T%C3%A4ufer%2C+M">Matthias T盲ufer</a>, <a href="/search/math-ph?searchtype=author&query=Wintermayr%2C+J">Jens Wintermayr</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2301.05076v1-abstract-short" style="display: inline;"> We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome lattice and the $(3.12)^2$ ``Super-Kagome'' lattice. We characterize all possible choices for edge wei… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.05076v1-abstract-full').style.display = 'inline'; document.getElementById('2301.05076v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2301.05076v1-abstract-full" style="display: none;"> We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome lattice and the $(3.12)^2$ ``Super-Kagome'' lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an ``all-or-nothing'' phenomenon in the sense that there is no perturbation which can destroy only one flat band while preserving the other. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.05076v1-abstract-full').style.display = 'none'; document.getElementById('2301.05076v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 6 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C90; 58J50; 81Q35; 92E10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ann. Henri Poincar茅 (2023) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.13954">arXiv:2212.13954</a> <span> [<a href="https://arxiv.org/pdf/2212.13954">pdf</a>, <a href="https://arxiv.org/ps/2212.13954">ps</a>, <a href="https://arxiv.org/format/2212.13954">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1090/proc/16578">10.1090/proc/16578 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Comparing the spectrum of Schr枚dinger operators on quantum graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bifulco%2C+P">Patrizio Bifulco</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.13954v1-abstract-short" style="display: inline;"> We study Schr枚dinger operators on compact finite metric graphs subject to $未$-coupling and standard boundary conditions. We compare the $n$-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of deviations. By doing this, we generalize recent results from Rudnick et al. obtained for domains in $\mathbb{R}^2$ to the setting of quantum graphs. This al… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.13954v1-abstract-full').style.display = 'inline'; document.getElementById('2212.13954v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.13954v1-abstract-full" style="display: none;"> We study Schr枚dinger operators on compact finite metric graphs subject to $未$-coupling and standard boundary conditions. We compare the $n$-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of deviations. By doing this, we generalize recent results from Rudnick et al. obtained for domains in $\mathbb{R}^2$ to the setting of quantum graphs. This also leads to a generalization of related results previously and independently obtained in [arXiv:2212.09143] and [arXiv:2212.12531] for metric graphs. In addition, based on our main result, we introduce some notions of circumference for a (quantum) graph which might prove useful in the future. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.13954v1-abstract-full').style.display = 'none'; document.getElementById('2212.13954v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L05; 81Q35; 34L15; 34L20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proc. Amer. Math. Soc. (2023) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.15110">arXiv:2110.15110</a> <span> [<a href="https://arxiv.org/pdf/2110.15110">pdf</a>, <a href="https://arxiv.org/ps/2110.15110">ps</a>, <a href="https://arxiv.org/format/2110.15110">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3233/ASY-221806">10.3233/ASY-221806 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the spectral gap of higher-dimensional Schr枚dinger operators on large domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=T%C3%A4ufer%2C+M">Matthias T盲ufer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2110.15110v1-abstract-short" style="display: inline;"> We study the asymptotic behaviour of the spectral gap of Schr枚dinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.15110v1-abstract-full" style="display: none;"> We study the asymptotic behaviour of the spectral gap of Schr枚dinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.15110v1-abstract-full').style.display = 'none'; document.getElementById('2110.15110v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L15; 34L40; 47E05; 47A10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Asymptotic Analysis (2022) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.04587">arXiv:2110.04587</a> <span> [<a href="https://arxiv.org/pdf/2110.04587">pdf</a>, <a href="https://arxiv.org/ps/2110.04587">ps</a>, <a href="https://arxiv.org/format/2110.04587">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/jpr.2022.54">10.1017/jpr.2022.54 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Bose--Einstein condensation for particles with repulsive short-range pair interactions in a Poisson random external potential in $\mathbb R^d$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pechmann%2C+M">Maximilian Pechmann</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2110.04587v1-abstract-short" style="display: inline;"> We study Bose gases in $d$ dimensions, $d \ge 2$, with short-range repulsive pair interactions, at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose--Einstein condensation into… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.04587v1-abstract-full').style.display = 'inline'; document.getElementById('2110.04587v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.04587v1-abstract-full" style="display: none;"> We study Bose gases in $d$ dimensions, $d \ge 2$, with short-range repulsive pair interactions, at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose--Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the eigenstates of the underlying one-particle Hamiltonian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.04587v1-abstract-full').style.display = 'none'; document.getElementById('2110.04587v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B44; 60G55; 81V70; 82B10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Applied Probability (2022) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.03813">arXiv:2103.03813</a> <span> [<a href="https://arxiv.org/pdf/2103.03813">pdf</a>, <a href="https://arxiv.org/ps/2103.03813">ps</a>, <a href="https://arxiv.org/format/2103.03813">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00013-022-01786-2">10.1007/s00013-022-01786-2 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A lower bound on the spectral gap of Schr枚dinger operators with weak potentials of compact support </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.03813v1-abstract-short" style="display: inline;"> In this paper we continue the study of the spectral gap of Schr枚dinger operators on large intervals and subject to Neumann boundary conditions. The main goal is to derive a lower bound on the spectral gap which is polynomial in the interval length. This bound is derived for a class of bounded potentials of compact support which are weak enough in a suitable sense. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.03813v1-abstract-full" style="display: none;"> In this paper we continue the study of the spectral gap of Schr枚dinger operators on large intervals and subject to Neumann boundary conditions. The main goal is to derive a lower bound on the spectral gap which is polynomial in the interval length. This bound is derived for a class of bounded potentials of compact support which are weak enough in a suitable sense. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.03813v1-abstract-full').style.display = 'none'; document.getElementById('2103.03813v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L15; 34L40; 47E05; 47A10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Archiv der Mathematik (2022) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.03816">arXiv:2102.03816</a> <span> [<a href="https://arxiv.org/pdf/2102.03816">pdf</a>, <a href="https://arxiv.org/ps/2102.03816">ps</a>, <a href="https://arxiv.org/format/2102.03816">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00013-022-01786-2">10.1007/s00013-022-01786-2 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A lower bound on the spectral gap of one-dimensional Schr枚dinger operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2102.03816v3-abstract-short" style="display: inline;"> In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schr枚dinger operators with non-negative bounded potentials and subject to Neumann boundary conditions. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.03816v3-abstract-full" style="display: none;"> In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schr枚dinger operators with non-negative bounded potentials and subject to Neumann boundary conditions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.03816v3-abstract-full').style.display = 'none'; document.getElementById('2102.03816v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Improved results; title and text amended accordingly; comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L15; 34L40; 47E05; 47A10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Archiv der Mathematik (2022) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.09060">arXiv:2012.09060</a> <span> [<a href="https://arxiv.org/pdf/2012.09060">pdf</a>, <a href="https://arxiv.org/ps/2012.09060">ps</a>, <a href="https://arxiv.org/format/2012.09060">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00013-024-02060-3">10.1007/s00013-024-02060-3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the spectral gap of one-dimensional Schr枚dinger operators on large intervals </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=T%C3%A4ufer%2C+M">Matthias T盲ufer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2012.09060v2-abstract-short" style="display: inline;"> We study the effect of non-negative potentials on the spectral gap of one-dimensional Schr枚dinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials which characterize its asymptotic behaviour. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.09060v2-abstract-full" style="display: none;"> We study the effect of non-negative potentials on the spectral gap of one-dimensional Schr枚dinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials which characterize its asymptotic behaviour. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.09060v2-abstract-full').style.display = 'none'; document.getElementById('2012.09060v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34L15; 34L40; 47E05; 47A10; </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Arch. Math. (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.06448">arXiv:2007.06448</a> <span> [<a href="https://arxiv.org/pdf/2007.06448">pdf</a>, <a href="https://arxiv.org/ps/2007.06448">ps</a>, <a href="https://arxiv.org/format/2007.06448">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1090/proc/15424">10.1090/proc/15424 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the effect of repulsive interactions on Bose-Einstein condensation in the Luttinger-Sy model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pechmann%2C+M">Maximilian Pechmann</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.06448v1-abstract-short" style="display: inline;"> In this paper we investigate the effect of repulsive pair interactions on Bose-Einstein condensation in a well-established random one-dimensional system known as the Luttinger-Sy model at positive temperature. We study separately hard core interactions as well as a class of more general repulsive interactions, also allowing for a scaling of certain interaction parameters in the thermodynamic limit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.06448v1-abstract-full').style.display = 'inline'; document.getElementById('2007.06448v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.06448v1-abstract-full" style="display: none;"> In this paper we investigate the effect of repulsive pair interactions on Bose-Einstein condensation in a well-established random one-dimensional system known as the Luttinger-Sy model at positive temperature. We study separately hard core interactions as well as a class of more general repulsive interactions, also allowing for a scaling of certain interaction parameters in the thermodynamic limit. As a main result, we prove in both cases that for sufficiently strong interactions all eigenstates of the non-interacting one-particle Luttinger-Sy Hamiltonian as well as any sufficiently localized one-particle state are almost surely not macroscopically occupied. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.06448v1-abstract-full').style.display = 'none'; document.getElementById('2007.06448v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B44; 81V70; 82B10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proc. Amer. Math. Soc. (2021) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.03200">arXiv:1911.03200</a> <span> [<a href="https://arxiv.org/pdf/1911.03200">pdf</a>, <a href="https://arxiv.org/ps/1911.03200">ps</a>, <a href="https://arxiv.org/format/1911.03200">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.matpur.2020.07.006">10.1016/j.matpur.2020.07.006 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On a condition for type-I Bose-Einstein condensation in random potentials in $d$ dimensions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pechmann%2C+M">Maximilian Pechmann</a>, <a href="/search/math-ph?searchtype=author&query=Spitzer%2C+W">Wolfgang Spitzer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1911.03200v1-abstract-short" style="display: inline;"> In this paper we discuss Bose-Einstein condensation (BEC) in systems of pairwise non-interacting bosons in random potentials in $d$ dimensions. Working in a rather general framework, we provide a "gap condition" which is sufficient to conclude existence of type-I BEC in probability and in the $r$th mean. We illustrate our results in the context of the well-known (one-dimensional) Luttinger-Sy mode… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.03200v1-abstract-full').style.display = 'inline'; document.getElementById('1911.03200v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.03200v1-abstract-full" style="display: none;"> In this paper we discuss Bose-Einstein condensation (BEC) in systems of pairwise non-interacting bosons in random potentials in $d$ dimensions. Working in a rather general framework, we provide a "gap condition" which is sufficient to conclude existence of type-I BEC in probability and in the $r$th mean. We illustrate our results in the context of the well-known (one-dimensional) Luttinger-Sy model. Here, whenever the particle density exceeds a critical value, we show in addition that only the ground state is macroscopically occupied. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.03200v1-abstract-full').style.display = 'none'; document.getElementById('1911.03200v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal de Math. Pures et Appli. (2020) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1908.02554">arXiv:1908.02554</a> <span> [<a href="https://arxiv.org/pdf/1908.02554">pdf</a>, <a href="https://arxiv.org/ps/1908.02554">ps</a>, <a href="https://arxiv.org/format/1908.02554">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11005-019-01246-z">10.1007/s11005-019-01246-z <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Attractive conical surfaces create infinitely many bound states </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Egger%2C+S">Sebastian Egger</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pankrashkin%2C+K">Konstantin Pankrashkin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1908.02554v1-abstract-short" style="display: inline;"> In this paper we study spectral properties of a three-dimensional Schr枚dinger operator $-螖+V$ with a potential $V$ given, modulo rapidly decaying terms, by a function of the distance of $x \in \mathbb{R}^3$ to an infinite conical hypersurface with a smooth cross-section. As a main result we show that there are infinitely many discrete eigenvalues accumulating at the bottom of the essential spectru… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.02554v1-abstract-full').style.display = 'inline'; document.getElementById('1908.02554v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1908.02554v1-abstract-full" style="display: none;"> In this paper we study spectral properties of a three-dimensional Schr枚dinger operator $-螖+V$ with a potential $V$ given, modulo rapidly decaying terms, by a function of the distance of $x \in \mathbb{R}^3$ to an infinite conical hypersurface with a smooth cross-section. As a main result we show that there are infinitely many discrete eigenvalues accumulating at the bottom of the essential spectrum which itself is identified as the ground-state energy of a certain one-dimensional operator. Most importantly, based on a result of Kirsch and Simon we are able to establish the asymptotic behavior of the eigenvalue counting function using an explicit spectral-geometric quantity associated with the cross-section. This shows a universal character of some previous results on conical layers and $未$-potentials created by conical surfaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.02554v1-abstract-full').style.display = 'none'; document.getElementById('1908.02554v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81Q10; 81Q80; 81Q37; 81Q35 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Lett. Math. Phys. 110 (2020) 945-968 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.11804">arXiv:1812.11804</a> <span> [<a href="https://arxiv.org/pdf/1812.11804">pdf</a>, <a href="https://arxiv.org/ps/1812.11804">ps</a>, <a href="https://arxiv.org/format/1812.11804">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.7153/oam-2020-14-45">10.7153/oam-2020-14-45 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the number of isolated eigenvalues of a pair of particles in a quantum wire </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.11804v1-abstract-short" style="display: inline;"> In this note we consider a pair of particles moving on the positive half-line with the pairing generated by a hard-wall potential. This model was first introduced in [arXiv:1604.06693] and later applied to investigate condensation of pairs of electrons in a quantum wire [arXiv:1708.03753,arXiv:1801.00696]. For this, a detailed spectral analysis proved necessary and as a part of this it was shown i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.11804v1-abstract-full').style.display = 'inline'; document.getElementById('1812.11804v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.11804v1-abstract-full" style="display: none;"> In this note we consider a pair of particles moving on the positive half-line with the pairing generated by a hard-wall potential. This model was first introduced in [arXiv:1604.06693] and later applied to investigate condensation of pairs of electrons in a quantum wire [arXiv:1708.03753,arXiv:1801.00696]. For this, a detailed spectral analysis proved necessary and as a part of this it was shown in [arXiv:1708.03753] that, in a special case, the discrete spectrum of the Hamiltonian consists of a single eigenvalue only. It is the aim of this note to prove that this is generally the case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.11804v1-abstract-full').style.display = 'none'; document.getElementById('1812.11804v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B26; 82B10; 81V70; 81Q10; 81Q35; 82D77; 82D80; 82D55 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Operators and Matrices (2020) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.06500">arXiv:1812.06500</a> <span> [<a href="https://arxiv.org/pdf/1812.06500">pdf</a>, <a href="https://arxiv.org/ps/1812.06500">ps</a>, <a href="https://arxiv.org/format/1812.06500">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4171/JST/331">10.4171/JST/331 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Bound states of a pair of particles on the half-line with a general interaction potential </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Egger%2C+S">Sebastian Egger</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pankrashkin%2C+K">Konstantin Pankrashkin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.06500v1-abstract-short" style="display: inline;"> In this paper we study an interacting two-particle system on the positive half-line. We focus on spectral properties of the Hamiltonian for a large class of two-particle potentials. We characterize the essential spectrum and prove, as a main result, the existence of eigenvalues below the bottom of it. We also prove that the discrete spectrum contains only finitely many eigenvalues. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.06500v1-abstract-full" style="display: none;"> In this paper we study an interacting two-particle system on the positive half-line. We focus on spectral properties of the Hamiltonian for a large class of two-particle potentials. We characterize the essential spectrum and prove, as a main result, the existence of eigenvalues below the bottom of it. We also prove that the discrete spectrum contains only finitely many eigenvalues. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.06500v1-abstract-full').style.display = 'none'; document.getElementById('1812.06500v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81Q10; 81Q80; 81Q37; 81Q35 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Spectr. Theory 10 (2020) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1808.10811">arXiv:1808.10811</a> <span> [<a href="https://arxiv.org/pdf/1808.10811">pdf</a>, <a href="https://arxiv.org/ps/1808.10811">ps</a>, <a href="https://arxiv.org/format/1808.10811">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10955-019-02240-4">10.1007/s10955-019-02240-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On Bose-Einstein condensation in the Luttinger-Sy model with finite interaction strength </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pechmann%2C+M">Maximilian Pechmann</a>, <a href="/search/math-ph?searchtype=author&query=Spitzer%2C+W">Wolfgang Spitzer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1808.10811v1-abstract-short" style="display: inline;"> We study Bose-Einstein condensation (BEC) in the Luttinger-Sy model. Here, Bose point particles in one spatial dimension do not interact with each other, but, through a positive (repulsive) point potential with impurities which are randomly located along the real line according to the points of a Poisson process. Our emphasis is on the case in which the interaction strength is not infinite. As a m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.10811v1-abstract-full').style.display = 'inline'; document.getElementById('1808.10811v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1808.10811v1-abstract-full" style="display: none;"> We study Bose-Einstein condensation (BEC) in the Luttinger-Sy model. Here, Bose point particles in one spatial dimension do not interact with each other, but, through a positive (repulsive) point potential with impurities which are randomly located along the real line according to the points of a Poisson process. Our emphasis is on the case in which the interaction strength is not infinite. As a main result, we prove that in thermal equilibrium the one-particle ground state is macroscopically occupied, provided that the particle density is larger than a critical one depending on the temperature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.10811v1-abstract-full').style.display = 'none'; document.getElementById('1808.10811v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B44; 81V70; 82B10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Statistical Physics (2019) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.05280">arXiv:1805.05280</a> <span> [<a href="https://arxiv.org/pdf/1805.05280">pdf</a>, <a href="https://arxiv.org/ps/1805.05280">ps</a>, <a href="https://arxiv.org/format/1805.05280">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00013-018-1243-4">10.1007/s00013-018-1243-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On Lennard-Jones-type potentials on the half-line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Gregorio%2C+F">Federica Gregorio</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1805.05280v2-abstract-short" style="display: inline;"> In this paper we study a particle under the influence of a Lennard-Jones potential moving in a simple quantum wire modelled by the positive half-line. Despite its physical significance, this potential is only rarely studied in the literature and due to its singularity at the origin it cannot be considered as a standard perturbation of the one-dimensional Laplacian. It is therefore our aim to provi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.05280v2-abstract-full').style.display = 'inline'; document.getElementById('1805.05280v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.05280v2-abstract-full" style="display: none;"> In this paper we study a particle under the influence of a Lennard-Jones potential moving in a simple quantum wire modelled by the positive half-line. Despite its physical significance, this potential is only rarely studied in the literature and due to its singularity at the origin it cannot be considered as a standard perturbation of the one-dimensional Laplacian. It is therefore our aim to provide a thorough description of the Hamiltonian in one dimension via the construction of a suitable quadratic form. Our results include a discussion of spectral and scattering properties which finally allows us to generalise some results from [Robinson1974] as well as [RadinSimon1978]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.05280v2-abstract-full').style.display = 'none'; document.getElementById('1805.05280v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">various typos corrected; to appear in Archiv der Mathematik</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81Q10; 81Q37; 81Q80; 81Q35; 47E05; 47D08 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Arch. Math. (2018) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.00725">arXiv:1805.00725</a> <span> [<a href="https://arxiv.org/pdf/1805.00725">pdf</a>, <a href="https://arxiv.org/ps/1805.00725">ps</a>, <a href="https://arxiv.org/format/1805.00725">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/978-3-030-44097-8">10.1007/978-3-030-44097-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Many-particle quantum graphs: A review </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bolte%2C+J">Jens Bolte</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1805.00725v1-abstract-short" style="display: inline;"> In this paper we review recent work that has been done on quantum many-particle systems on metric graphs. Topics include the implementation of singular interactions, Bose-Einstein condensation, sovable models and spectral properties of some simple models in connection with superconductivity in wires. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.00725v1-abstract-full" style="display: none;"> In this paper we review recent work that has been done on quantum many-particle systems on metric graphs. Topics include the implementation of singular interactions, Bose-Einstein condensation, sovable models and spectral properties of some simple models in connection with superconductivity in wires. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.00725v1-abstract-full').style.display = 'none'; document.getElementById('1805.00725v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Discrete and Continuous Models in the Theory of Networks (2020); Birkh盲user: Operator Theory: Advances and Applications </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1804.07697">arXiv:1804.07697</a> <span> [<a href="https://arxiv.org/pdf/1804.07697">pdf</a>, <a href="https://arxiv.org/ps/1804.07697">ps</a>, <a href="https://arxiv.org/format/1804.07697">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00023-019-00771-w">10.1007/s00023-019-00771-w <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Bose--Einstein condensation in the Luttinger--Sy model with contact interaction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=Pechmann%2C+M">Maximilian Pechmann</a>, <a href="/search/math-ph?searchtype=author&query=Spitzer%2C+W">Wolfgang Spitzer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1804.07697v1-abstract-short" style="display: inline;"> We study bosons on the real line in a Poisson random potential (Luttinger--Sy model) with contact interaction in the thermodynamic limit at absolute zero temperature. We prove that generalized Bose--Einstein condensation (BEC) occurs almost surely if the intensity $谓_N$ of the Poisson potential satisfies $[\ln (N)]^4/N^{1 - 2畏} \ll 谓_N \lesssim 1$ for arbitrary $0 < 畏\leq 1/3$. We also show that t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.07697v1-abstract-full').style.display = 'inline'; document.getElementById('1804.07697v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1804.07697v1-abstract-full" style="display: none;"> We study bosons on the real line in a Poisson random potential (Luttinger--Sy model) with contact interaction in the thermodynamic limit at absolute zero temperature. We prove that generalized Bose--Einstein condensation (BEC) occurs almost surely if the intensity $谓_N$ of the Poisson potential satisfies $[\ln (N)]^4/N^{1 - 2畏} \ll 谓_N \lesssim 1$ for arbitrary $0 < 畏\leq 1/3$. We also show that the contact interaction alters the type of condensation, going from a type-I BEC to a type-III BEC as the strength of this interaction is increased. Furthermore, for sufficiently strong contact interactions and $0 < 畏< 1/6$ we prove that the mean particle density in the largest interval is almost surely bounded asymptotically by $谓_NN^{3/5+未}$ for $未> 0$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.07697v1-abstract-full').style.display = 'none'; document.getElementById('1804.07697v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B44; 81V70; 82B10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Annales Henri Poincar茅 (2019) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.00696">arXiv:1801.00696</a> <span> [<a href="https://arxiv.org/pdf/1801.00696">pdf</a>, <a href="https://arxiv.org/ps/1801.00696">ps</a>, <a href="https://arxiv.org/format/1801.00696">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.5026353">10.1063/1.5026353 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On pairs of interacting electrons in a quantum wire </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1801.00696v1-abstract-short" style="display: inline;"> In this paper we consider pairs of interacting electrons moving in a simple quantum wire, namely the half-line. In particular, we extend the results obtained in [arXiv:1708.03753] by allowing for contact interactions of the Lieb-Liniger type between the two electrons constituting the pair. We construct the associated Hamiltonian rigorously and study its spectral properties. We then investigate Bos… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.00696v1-abstract-full').style.display = 'inline'; document.getElementById('1801.00696v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.00696v1-abstract-full" style="display: none;"> In this paper we consider pairs of interacting electrons moving in a simple quantum wire, namely the half-line. In particular, we extend the results obtained in [arXiv:1708.03753] by allowing for contact interactions of the Lieb-Liniger type between the two electrons constituting the pair. We construct the associated Hamiltonian rigorously and study its spectral properties. We then investigate Bose-Einstein condensation of pairs and prove, as a main result, the existence of condensation whenever the Hamiltonian has a non-trivial discrete spectrum. Most importantly, condensation is proved for very weak and very strong contact interactions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.00696v1-abstract-full').style.display = 'none'; document.getElementById('1801.00696v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B26; 82B10; 81V70; 81Q10; 81Q35; 82D77; 82D80; 82D55 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Mathematical Physics 59, 063504 (2018) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1712.07650">arXiv:1712.07650</a> <span> [<a href="https://arxiv.org/pdf/1712.07650">pdf</a>, <a href="https://arxiv.org/ps/1712.07650">ps</a>, <a href="https://arxiv.org/format/1712.07650">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4213/tmf9720">10.4213/tmf9720 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the impact of surface defects on a condensate of electron pairs in a quantum wire </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1712.07650v3-abstract-short" style="display: inline;"> In this paper we are interested in understanding the impact of surface defects on a condensate of electron pairs in a quantum wire. Based on previous results we establish a simple mathematical model in order to account for such surface effects. For a system of non-interacting pairs, we will prove the destruction of the condensate in the bulk. Finally, taking repulsive interactions between the pair… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.07650v3-abstract-full').style.display = 'inline'; document.getElementById('1712.07650v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1712.07650v3-abstract-full" style="display: none;"> In this paper we are interested in understanding the impact of surface defects on a condensate of electron pairs in a quantum wire. Based on previous results we establish a simple mathematical model in order to account for such surface effects. For a system of non-interacting pairs, we will prove the destruction of the condensate in the bulk. Finally, taking repulsive interactions between the pairs into account, we will show that the condensate is recovered for pair densities larger than a critical one given the number of the surface defects is not too large. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.07650v3-abstract-full').style.display = 'none'; document.getElementById('1712.07650v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Title changed; improved proofs</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B26; 82B10; 81V70; 81Q10; 81Q35; 82D77; 82D80; 82D55 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Theoretical and Mathematical Physics, 2020, Volume 203 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1708.03753">arXiv:1708.03753</a> <span> [<a href="https://arxiv.org/pdf/1708.03753">pdf</a>, <a href="https://arxiv.org/ps/1708.03753">ps</a>, <a href="https://arxiv.org/format/1708.03753">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0034-4877(19)30028-X">10.1016/S0034-4877(19)30028-X <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On bound electron pairs in a quantum wire </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1708.03753v2-abstract-short" style="display: inline;"> Based on the quantum two-body problem introduced in [arXiv:1604.06693] we consider bound pairs of electrons moving on the positive half-line. The analysis is motivated by the ground-breaking work of Cooper who identified the pairing of electrons as a possible explanation for superconducting behaviour in metals. In this paper we are interested in the connection between the topologies of the underly… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.03753v2-abstract-full').style.display = 'inline'; document.getElementById('1708.03753v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1708.03753v2-abstract-full" style="display: none;"> Based on the quantum two-body problem introduced in [arXiv:1604.06693] we consider bound pairs of electrons moving on the positive half-line. The analysis is motivated by the ground-breaking work of Cooper who identified the pairing of electrons as a possible explanation for superconducting behaviour in metals. In this paper we are interested in the connection between the topologies of the underlying one- and two-particle configuration spaces and spectral properties of the Hamiltonian linked to a condensation of pairs. We derive explicit estimates for the energy gap and prove condensation for a gas of non-interacting pairs. Finally, we add some disorder to the system and prove destruction of the condensate. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.03753v2-abstract-full').style.display = 'none'; document.getElementById('1708.03753v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 August, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Improved proofs</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B26; 82B10; 81V70; 81Q10; 81Q35; 82D77; 82D80; 82D55 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Reports on Mathematical Physics (2019) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.00961">arXiv:1707.00961</a> <span> [<a href="https://arxiv.org/pdf/1707.00961">pdf</a>, <a href="https://arxiv.org/ps/1707.00961">ps</a>, <a href="https://arxiv.org/format/1707.00961">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00013-018-1292-8">10.1007/s00013-018-1292-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A remark on the effect of random singular two-particle interactions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1707.00961v1-abstract-short" style="display: inline;"> In this note we study a two-particle bound system (molecule) moving on the positive half-line under the influence of randomly distributed singular two-particle interactions generated by a Poisson process. We give a rigorous definition of the underlying Hamiltonian and study its spectral properties. As a main result we prove that, with finite probability, the random interactions destroy the discret… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.00961v1-abstract-full').style.display = 'inline'; document.getElementById('1707.00961v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.00961v1-abstract-full" style="display: none;"> In this note we study a two-particle bound system (molecule) moving on the positive half-line under the influence of randomly distributed singular two-particle interactions generated by a Poisson process. We give a rigorous definition of the underlying Hamiltonian and study its spectral properties. As a main result we prove that, with finite probability, the random interactions destroy the discrete part of the spectrum which is present in the free system. Most interestingly, this phenomenon is somewhat contrary to the role attributed to random interactions in the context of Anderson localisation where disorder is generally associated with a suppression of transport. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.00961v1-abstract-full').style.display = 'none'; document.getElementById('1707.00961v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81V70; 81Q80; 81Q35; 81Q10; 81Q50; 81Q37; </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Archiv der Mathematik (2019) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.00851">arXiv:1702.00851</a> <span> [<a href="https://arxiv.org/pdf/1702.00851">pdf</a>, <a href="https://arxiv.org/ps/1702.00851">ps</a>, <a href="https://arxiv.org/format/1702.00851">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0129055X17500325">10.1142/S0129055X17500325 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Scattering properties of two singularly interacting particles on the half-line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Egger%2C+S">Sebastian Egger</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.00851v2-abstract-short" style="display: inline;"> We analyze scattering in a system of two (distinguishable) particles moving on the half-line $\overline{\rz}_+$ under the influence of singular two-particle interactions. Most importantly, due to the spatial localization of the interactions the two-body problem is of a non-separable nature. We will discuss the presence of embedded eigenvalues and using the obtained knowledge about the kernel of th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.00851v2-abstract-full').style.display = 'inline'; document.getElementById('1702.00851v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.00851v2-abstract-full" style="display: none;"> We analyze scattering in a system of two (distinguishable) particles moving on the half-line $\overline{\rz}_+$ under the influence of singular two-particle interactions. Most importantly, due to the spatial localization of the interactions the two-body problem is of a non-separable nature. We will discuss the presence of embedded eigenvalues and using the obtained knowledge about the kernel of the resolvent we prove a version of the limiting absorption principle. Furthermore, by an appropriate adaptation of the Lippmann-Schwinger approach we are able to construct generalized eigenfunctions which consequently allow us to establish an explicit expression for the (on-shell) scattering amplitude. An approximation of the scattering amplitude in the weak-coupling limit is also derived. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.00851v2-abstract-full').style.display = 'none'; document.getElementById('1702.00851v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81U05; 81U15; 81Q10; 81Q50; 81Q37; 81Q35; 81Q80 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Rev. Math. Phys. 29 (2017), no. 10, 1750032 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1604.06693">arXiv:1604.06693</a> <span> [<a href="https://arxiv.org/pdf/1604.06693">pdf</a>, <a href="https://arxiv.org/format/1604.06693">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Superconductivity">cond-mat.supr-con</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0034-4877(17)30068-X">10.1016/S0034-4877(17)30068-X <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On a two-particle bound system on the half-line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=M%C3%BChlenbruch%2C+T">Tobias M眉hlenbruch</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1604.06693v2-abstract-short" style="display: inline;"> In this paper we provide an extension of the model discussed in [arXiv:1504.08283] describing two singularly interacting particles on the half-line. In this model, the particles are interacting only whenever at least one particle is situated at the origin. Stimulated by [arXiv:1503.08814] we then provide a generalisation of this model in order to include additional interactions between the particl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.06693v2-abstract-full').style.display = 'inline'; document.getElementById('1604.06693v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1604.06693v2-abstract-full" style="display: none;"> In this paper we provide an extension of the model discussed in [arXiv:1504.08283] describing two singularly interacting particles on the half-line. In this model, the particles are interacting only whenever at least one particle is situated at the origin. Stimulated by [arXiv:1503.08814] we then provide a generalisation of this model in order to include additional interactions between the particles leading to a molecular-like state. We give a precise mathematical formulation of the Hamiltonian of the system and perform spectral analysis. In particular, we are interested in the effect of the singular two-particle interactions onto the molecule. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.06693v2-abstract-full').style.display = 'none'; document.getElementById('1604.06693v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 June, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81Q35 (Primary); 81Q80; 81Q10; 82D55; 82D77; 35J25 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Rep. Math. Phys. 80 (2017), no. 2 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1504.08283">arXiv:1504.08283</a> <span> [<a href="https://arxiv.org/pdf/1504.08283">pdf</a>, <a href="https://arxiv.org/ps/1504.08283">ps</a>, <a href="https://arxiv.org/format/1504.08283">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.4940698">10.1063/1.4940698 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Two interacting particles on the half-line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a>, <a href="/search/math-ph?searchtype=author&query=M%C3%BChlenbruch%2C+T">Tobias M眉hlenbruch</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1504.08283v2-abstract-short" style="display: inline;"> In the case of compact quantum graphs, many-particle models with singular two-particle interactions where introduced in [arXiv:1207.5648, arXiv:1112.4751] to provide a paradigm for further studies on many-particle quantum chaos. In this note, we discuss various aspects of such singular interactions in a two-particle system restricted to the half-line $\mathbb{R}_+$. Among others, we give a descrip… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.08283v2-abstract-full').style.display = 'inline'; document.getElementById('1504.08283v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1504.08283v2-abstract-full" style="display: none;"> In the case of compact quantum graphs, many-particle models with singular two-particle interactions where introduced in [arXiv:1207.5648, arXiv:1112.4751] to provide a paradigm for further studies on many-particle quantum chaos. In this note, we discuss various aspects of such singular interactions in a two-particle system restricted to the half-line $\mathbb{R}_+$. Among others, we give a description of the spectrum of the two-particle Hamiltonian and obtain upper bounds on the number of eigenstates below the essential spectrum. We also specify conditions under which there is exactly one such eigenstate. As a final result, it is shown that the ground state is unique and decays exponentially as $\sqrt{x^2+y^2} \to \infty$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.08283v2-abstract-full').style.display = 'none'; document.getElementById('1504.08283v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81Q35 (Primary); 35J25; 35P15; 81Q10; 81V70 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Mathematical Physics 57, 023509 (2016) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1411.7330">arXiv:1411.7330</a> <span> [<a href="https://arxiv.org/pdf/1411.7330">pdf</a>, <a href="https://arxiv.org/ps/1411.7330">ps</a>, <a href="https://arxiv.org/format/1411.7330">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.4946044">10.1063/1.4946044 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Instability of Bose-Einstein condensation on quantum graphs under repulsive perturbations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bolte%2C+J">Jens Bolte</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1411.7330v1-abstract-short" style="display: inline;"> In this Note we investigate Bose-Einstein condensation in interacting quantum many-particle systems on graphs. We extend previous results obtained for particles on an interval and show that even arbitrarily small repulsive two-particle interactions destroy a condensate in the non-interacting Bose gas. Our results also cover singular two-particle interactions, such as the well-known Lieb-Lininger m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.7330v1-abstract-full').style.display = 'inline'; document.getElementById('1411.7330v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1411.7330v1-abstract-full" style="display: none;"> In this Note we investigate Bose-Einstein condensation in interacting quantum many-particle systems on graphs. We extend previous results obtained for particles on an interval and show that even arbitrarily small repulsive two-particle interactions destroy a condensate in the non-interacting Bose gas. Our results also cover singular two-particle interactions, such as the well-known Lieb-Lininger model, in the thermodynamic limit. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.7330v1-abstract-full').style.display = 'none'; document.getElementById('1411.7330v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 November, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82D50; 81V70; 46N50 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Mathematical Physics 57, 043301 (2016) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1403.0271">arXiv:1403.0271</a> <span> [<a href="https://arxiv.org/pdf/1403.0271">pdf</a>, <a href="https://arxiv.org/ps/1403.0271">ps</a>, <a href="https://arxiv.org/format/1403.0271">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/9789814618144_0016">10.1142/9789814618144_0016 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Bose-Einstein condensation on quantum graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bolte%2C+J">Jens Bolte</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1403.0271v1-abstract-short" style="display: inline;"> We present results on Bose-Einstein condensation (BEC) on general compact quantum graphs, i.e., one-dimensional systems with a (potentially) complex topology. We first investigate non-interacting many-particle systems and provide a complete classification of systems that exhibit condensation. We then consider models with interactions that consist of a singular part as well as a hardcore part. In t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.0271v1-abstract-full').style.display = 'inline'; document.getElementById('1403.0271v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1403.0271v1-abstract-full" style="display: none;"> We present results on Bose-Einstein condensation (BEC) on general compact quantum graphs, i.e., one-dimensional systems with a (potentially) complex topology. We first investigate non-interacting many-particle systems and provide a complete classification of systems that exhibit condensation. We then consider models with interactions that consist of a singular part as well as a hardcore part. In this way we obtain generalisations of the Tonks-Girardeau gas to graphs. For this we find an absence of phase transitions which then indicates an absence of BEC. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.0271v1-abstract-full').style.display = 'none'; document.getElementById('1403.0271v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 March, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 pages, contribution to the proceedings of QMATH12</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mathematical results in quantum mechanics, World Sci. Publ., Hackensack, NJ, 2015 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1309.6091">arXiv:1309.6091</a> <span> [<a href="https://arxiv.org/pdf/1309.6091">pdf</a>, <a href="https://arxiv.org/ps/1309.6091">ps</a>, <a href="https://arxiv.org/format/1309.6091">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.4879497">10.1063/1.4879497 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Many-particle quantum graphs and Bose-Einstein condensation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bolte%2C+J">Jens Bolte</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1309.6091v1-abstract-short" style="display: inline;"> In this paper we propose quantum graphs as one-dimensional models with a complex topology to study Bose-Einstein condensation and phase transitions in a rigorous way. We fist investigate non-interacting many-particle systems on quantum graphs and provide a complete classification of systems that exhibit Bose-Einstein condensation. We then consider models of interacting particles that can be regard… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.6091v1-abstract-full').style.display = 'inline'; document.getElementById('1309.6091v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1309.6091v1-abstract-full" style="display: none;"> In this paper we propose quantum graphs as one-dimensional models with a complex topology to study Bose-Einstein condensation and phase transitions in a rigorous way. We fist investigate non-interacting many-particle systems on quantum graphs and provide a complete classification of systems that exhibit Bose-Einstein condensation. We then consider models of interacting particles that can be regarded as a generalisation of the well-known Tonks-Girardeau gas. Here our principal result is that no phase transitions occur in bosonic systems with repulsive hardcore interactions, indicating an absence of Bose-Einstein condensation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.6091v1-abstract-full').style.display = 'none'; document.getElementById('1309.6091v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 September, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Mathematical Physics 55, 061901 (2014) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1207.5648">arXiv:1207.5648</a> <span> [<a href="https://arxiv.org/pdf/1207.5648">pdf</a>, <a href="https://arxiv.org/ps/1207.5648">ps</a>, <a href="https://arxiv.org/format/1207.5648">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8113/46/4/045207">10.1088/1751-8113/46/4/045207 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum graphs with two-particle contact interactions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bolte%2C+J">Jens Bolte</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1207.5648v1-abstract-short" style="display: inline;"> We construct models of many-particle quantum graphs with singular two-particle contact interactions, which can be either hardcore- or delta-interactions. Self-adjoint realisations of the two-particle Laplacian including such interactions are obtained via their associated quadratic forms. We prove discreteness of spectra as well as Weyl laws for the asymptotic eigenvalue counts. These constructions… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1207.5648v1-abstract-full').style.display = 'inline'; document.getElementById('1207.5648v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1207.5648v1-abstract-full" style="display: none;"> We construct models of many-particle quantum graphs with singular two-particle contact interactions, which can be either hardcore- or delta-interactions. Self-adjoint realisations of the two-particle Laplacian including such interactions are obtained via their associated quadratic forms. We prove discreteness of spectra as well as Weyl laws for the asymptotic eigenvalue counts. These constructions are first performed for two distinguishable particles and then for two identicle bosons. Furthermore, we extend the models to N bosons with two-particle interactions, thus implementing the Lieb-Liniger model on a graph. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1207.5648v1-abstract-full').style.display = 'none'; document.getElementById('1207.5648v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 July, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Mth. Theor. 46 (2013) 045207 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1112.4751">arXiv:1112.4751</a> <span> [<a href="https://arxiv.org/pdf/1112.4751">pdf</a>, <a href="https://arxiv.org/ps/1112.4751">ps</a>, <a href="https://arxiv.org/format/1112.4751">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8113/46/4/045206">10.1088/1751-8113/46/4/045206 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum graphs with singular two-particle interactions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bolte%2C+J">Jens Bolte</a>, <a href="/search/math-ph?searchtype=author&query=Kerner%2C+J">Joachim Kerner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1112.4751v1-abstract-short" style="display: inline;"> We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions. In order to find such Hamiltonians closed and semi-bounded quadratic forms are constructed, from which the ass… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.4751v1-abstract-full').style.display = 'inline'; document.getElementById('1112.4751v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1112.4751v1-abstract-full" style="display: none;"> We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions. In order to find such Hamiltonians closed and semi-bounded quadratic forms are constructed, from which the associated self-adjoint operators are extracted. We provide a general characterisation of such operators and, furthermore, produce certain classes of examples. We then consider identical particles and project to the bosonic and fermionic subspaces. Finally, we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate Weyl asymptotic law. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.4751v1-abstract-full').style.display = 'none'; document.getElementById('1112.4751v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 46 (2013) 045206 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

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