S. Fournais, L. Junge, T. Girardot, L. Morin, M. Olivieri, A. Triay
We consider a dilute Bose gas in the thermodynamic limit and prove a lower bound on the free energy for low temperatures which is in agreement with the conjecture of Lee-Huang-Yang on the excitation spectrum of the system. Combining techniques of cite{FS2} and cite{HHNST}, we give a simpler and shorter proof resolving the case of strong interactions, including the hard-core potential.
{"title":"The free energy of dilute Bose gases at low temperatures interacting via strong potentials","authors":"S. Fournais, L. Junge, T. Girardot, L. Morin, M. Olivieri, A. Triay","doi":"arxiv-2408.14222","DOIUrl":"https://doi.org/arxiv-2408.14222","url":null,"abstract":"We consider a dilute Bose gas in the thermodynamic limit and prove a lower\u0000bound on the free energy for low temperatures which is in agreement with the\u0000conjecture of Lee-Huang-Yang on the excitation spectrum of the system.\u0000Combining techniques of cite{FS2} and cite{HHNST}, we give a simpler and\u0000shorter proof resolving the case of strong interactions, including the\u0000hard-core potential.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we provide an affirmative answer to [16, Conjecture 1.5] on the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary hypersurfaces in the Euclidean half-space. More generally, we establish a theory for capillary convex bodies in the half-space and prove a general Alexandrov-Fenchel inequality for mixed volumes of capillary convex bodies. The conjecture [16, Conjecture 1.5] follows as its consequence.
{"title":"Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary. II","authors":"Xinqun Mei, Guofang Wang, Liangjun Weng, Chao Xia","doi":"arxiv-2408.13655","DOIUrl":"https://doi.org/arxiv-2408.13655","url":null,"abstract":"In this paper, we provide an affirmative answer to [16, Conjecture 1.5] on\u0000the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary\u0000hypersurfaces in the Euclidean half-space. More generally, we establish a\u0000theory for capillary convex bodies in the half-space and prove a general\u0000Alexandrov-Fenchel inequality for mixed volumes of capillary convex bodies. The\u0000conjecture [16, Conjecture 1.5] follows as its consequence.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"172 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we show that for a family of quantum walk models with electric fields, the spectrum is the unit circle for any irrational field. The result also holds for the associated CMV matrices defined by skew-shifts. Generalizations to CMV matrices with skew-shifts on higher dimensional torus are also obtained.
{"title":"On the spectrum of electric quantum walk and related CMV matrices","authors":"Fan Yang","doi":"arxiv-2408.12724","DOIUrl":"https://doi.org/arxiv-2408.12724","url":null,"abstract":"In this note, we show that for a family of quantum walk models with electric\u0000fields, the spectrum is the unit circle for any irrational field. The result\u0000also holds for the associated CMV matrices defined by skew-shifts.\u0000Generalizations to CMV matrices with skew-shifts on higher dimensional torus\u0000are also obtained.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove existence of such eigenvalues, estimate how many eigenvalues there are, and give upper and lower bounds for them.
{"title":"On the existence of eigenvalues of a one-dimensional Dirac operator","authors":"Daniel Sánchez-Mendoza, Monika Winklmeier","doi":"arxiv-2408.12697","DOIUrl":"https://doi.org/arxiv-2408.12697","url":null,"abstract":"The aim of this paper is to study the existence of eigenvalues in the gap of\u0000the essential spectrum of the one-dimensional Dirac operator in the presence of\u0000a bounded potential. We employ a generalized variational principle to prove\u0000existence of such eigenvalues, estimate how many eigenvalues there are, and\u0000give upper and lower bounds for them.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide inequalities enabling to bound the error between the exact solution and an approximated solution of an eigenvalue problem, obtained by subspace projection, as in the reduced basis method. We treat self-adjoint operators and degenerate cases. We apply the bounds to the eigenvector continuation method, which consists in creating the reduced space by using basis vectors extracted from perturbation theory.
{"title":"On reduced basis methods for eigenvalue problems, with an application to eigenvector continuation","authors":"Louis Garrigue, Benjamin Stamm","doi":"arxiv-2408.11924","DOIUrl":"https://doi.org/arxiv-2408.11924","url":null,"abstract":"We provide inequalities enabling to bound the error between the exact\u0000solution and an approximated solution of an eigenvalue problem, obtained by\u0000subspace projection, as in the reduced basis method. We treat self-adjoint\u0000operators and degenerate cases. We apply the bounds to the eigenvector\u0000continuation method, which consists in creating the reduced space by using\u0000basis vectors extracted from perturbation theory.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we consider achieving the largest principle eigenvalue of a Robin Laplacian on a bounded domain $Omega$ by optimizing the Robin parameter function under an integral constraint. The main novelty of our approach lies in establishing a close relation between the problem under consideration and the asymptotic behavior of the Dirichlet heat content of $Omega$. By using this relation we deduce a two-term asymptotic expansion of the principle eigenvalue and discuss several applications.
{"title":"Optimizing the ground of a Robin Laplacian: asymptotic behavior","authors":"Pavel Exner, Hynek Kovarik","doi":"arxiv-2408.11636","DOIUrl":"https://doi.org/arxiv-2408.11636","url":null,"abstract":"In this note we consider achieving the largest principle eigenvalue of a\u0000Robin Laplacian on a bounded domain $Omega$ by optimizing the Robin parameter\u0000function under an integral constraint. The main novelty of our approach lies in\u0000establishing a close relation between the problem under consideration and the\u0000asymptotic behavior of the Dirichlet heat content of $Omega$. By using this\u0000relation we deduce a two-term asymptotic expansion of the principle eigenvalue\u0000and discuss several applications.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is an upgraded version of von Neumann's famous theory on self-adjoint extensions of symmetric operators. As implied in the title, we have incorporated complex analysis (and complex geometry) into this theory in an essential way. The roles played by Hermtian symmetric spaces and modern value distribution theory in the theory are clarified. In doing so, many new concepts are introduced and many new results are obtained.
{"title":"Complex analysis of symmetric operators, I","authors":"Yicao Wang","doi":"arxiv-2408.10968","DOIUrl":"https://doi.org/arxiv-2408.10968","url":null,"abstract":"This is an upgraded version of von Neumann's famous theory on self-adjoint\u0000extensions of symmetric operators. As implied in the title, we have\u0000incorporated complex analysis (and complex geometry) into this theory in an\u0000essential way. The roles played by Hermtian symmetric spaces and modern value\u0000distribution theory in the theory are clarified. In doing so, many new concepts\u0000are introduced and many new results are obtained.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Badreddine Benhellal, Miguel Camarasa, Konstantin Pankrashkin
In a recent paper Behrndt, Holzmann, and Stenzel introduced a new class of two-dimensional Schr"odinger operators with oblique transmissions along smooth curves. We extend most components of this analysis to the case of Lipschitz curves.
{"title":"On Schrödinger operators with oblique transmission conditions on non-smooth curves","authors":"Badreddine Benhellal, Miguel Camarasa, Konstantin Pankrashkin","doi":"arxiv-2408.09813","DOIUrl":"https://doi.org/arxiv-2408.09813","url":null,"abstract":"In a recent paper Behrndt, Holzmann, and Stenzel introduced a new class of\u0000two-dimensional Schr\"odinger operators with oblique transmissions along smooth\u0000curves. We extend most components of this analysis to the case of Lipschitz\u0000curves.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the eigenvalues of a one-dimensional semiclassical Schr"odinger operator, where the potential consist of two quadratic ends (that is, looks like a harmonic oscillator at each infinite end), possibly with a flat region in the middle. Such a potential notably has a discontinuity in the second derivative. We derive an asymptotic expansion, valid either in the high energy regime or the semiclassical regime, with a leading order term given by the Bohr-Sommerfeld quantization condition, and an asymptotic expansion consisting of negative powers of the leading order term, with coefficients that are oscillatory in the leading order term. We apply this expansion to study the results of the Gutzwiller Trace formula and the heat kernel asymptotic for this class of potentials, giving an idea into what results to expect for such trace formulas for non-smooth potentials.
{"title":"Asymptotic Expansion of the Eigenvalues of a Bathtub Potential with Quadratic Ends","authors":"Yuzhou Zou","doi":"arxiv-2408.09816","DOIUrl":"https://doi.org/arxiv-2408.09816","url":null,"abstract":"We consider the eigenvalues of a one-dimensional semiclassical Schr\"odinger\u0000operator, where the potential consist of two quadratic ends (that is, looks\u0000like a harmonic oscillator at each infinite end), possibly with a flat region\u0000in the middle. Such a potential notably has a discontinuity in the second\u0000derivative. We derive an asymptotic expansion, valid either in the high energy\u0000regime or the semiclassical regime, with a leading order term given by the\u0000Bohr-Sommerfeld quantization condition, and an asymptotic expansion consisting\u0000of negative powers of the leading order term, with coefficients that are\u0000oscillatory in the leading order term. We apply this expansion to study the\u0000results of the Gutzwiller Trace formula and the heat kernel asymptotic for this\u0000class of potentials, giving an idea into what results to expect for such trace\u0000formulas for non-smooth potentials.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper shows that the wave trace of a bounded and strictly convex planar domain may be arbitrarily smooth in a neighborhood of some point in the length spectrum. In other words, the Poisson relation, which asserts that the singular support of the wave trace is contained in the closure of $pm$ the length spectrum, can almost be made into a strict inclusion. To do so, we construct large families of domains for which there exist multiple periodic billiard orbits having the same length but different Maslov indices. Using the microlocal Balian-Bloch-Zelditch parametrix for wave invariants developed in our previous paper, we solve a large system of equations for the boundary curvature jets, which leads to the required cancellations. We call such periodic orbits silent, since they are undetectable from the ostensibly audible wave trace. Such cancellations show that there are potential limitations in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different mathematical objects, at least insofar as the wave trace is concerned.
{"title":"Silent Orbits and Cancellations in the Wave Trace","authors":"Illya Koval, Amir Vig","doi":"arxiv-2408.09238","DOIUrl":"https://doi.org/arxiv-2408.09238","url":null,"abstract":"This paper shows that the wave trace of a bounded and strictly convex planar\u0000domain may be arbitrarily smooth in a neighborhood of some point in the length\u0000spectrum. In other words, the Poisson relation, which asserts that the singular\u0000support of the wave trace is contained in the closure of $pm$ the length\u0000spectrum, can almost be made into a strict inclusion. To do so, we construct\u0000large families of domains for which there exist multiple periodic billiard\u0000orbits having the same length but different Maslov indices. Using the\u0000microlocal Balian-Bloch-Zelditch parametrix for wave invariants developed in\u0000our previous paper, we solve a large system of equations for the boundary\u0000curvature jets, which leads to the required cancellations. We call such\u0000periodic orbits silent, since they are undetectable from the ostensibly audible\u0000wave trace. Such cancellations show that there are potential limitations in\u0000using the wave trace for inverse spectral problems and more fundamentally, that\u0000the Laplace spectrum and length spectrum are inherently different mathematical\u0000objects, at least insofar as the wave trace is concerned.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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