We study analytic torsion and eta like invariants on CR contact manifolds of any dimension admitting a circle transverse action, and equipped with a unitary representation. We show that, when defined using the spectrum of relevant operators arising in this geometry, the spectral series involved can been interpreted in their whole, both from a topological viewpoint, and as purely dynamical functions of the Reeb flow.
{"title":"Topological and dynamical aspects of some spectral invariants of contact manifolds with circle action","authors":"Michel RuminLMO","doi":"arxiv-2409.11787","DOIUrl":"https://doi.org/arxiv-2409.11787","url":null,"abstract":"<div><p>We study analytic torsion and eta like invariants on CR contact\u0000manifolds of any dimension admitting a circle transverse action, and equipped\u0000with a unitary representation. We show that, when defined using the spectrum of\u0000relevant operators arising in this geometry, the spectral series involved can\u0000been interpreted in their whole, both from a topological viewpoint, and as\u0000purely dynamical functions of the Reeb flow.</p></div>","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"190 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248715","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 establish global bounds for solutions to stationary and time-dependent Schr"odinger equations associated with the sublaplacian $mathcal L$ on the Heisenberg group, as well as its pure fractional power $mathcal L^s$ and conformally invariant fractional power $mathcal L_s$. The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators -- a variant of Mourre's commutator method -- and Hardy's type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schr"odinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case $s=1$, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results.
{"title":"Uniform resolvent estimates, smoothing effects and spectral stability for the Heisenberg sublaplacian","authors":"Luca Fanelli, Haruya Mizutani, Luz Roncal, Nico Michele Schiavone","doi":"arxiv-2409.11943","DOIUrl":"https://doi.org/arxiv-2409.11943","url":null,"abstract":"We establish global bounds for solutions to stationary and time-dependent\u0000Schr\"odinger equations associated with the sublaplacian $mathcal L$ on the\u0000Heisenberg group, as well as its pure fractional power $mathcal L^s$ and\u0000conformally invariant fractional power $mathcal L_s$. The main ingredient is a\u0000new abstract uniform weighted resolvent estimate which is proved by using the\u0000method of weakly conjugate operators -- a variant of Mourre's commutator method\u0000-- and Hardy's type inequalities on the Heisenberg group. As applications, we\u0000show Kato-type smoothing effects for the time-dependent Schr\"odinger equation,\u0000and spectral stability of the sublaplacian perturbed by complex-valued decaying\u0000potentials satisfying an explicit subordination condition. In the local case\u0000$s=1$, we obtain uniform estimates without any symmetry or derivative loss,\u0000which improve previous results.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248709","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}
Ram Band, Siegfried Beckus, Barak Biber, Laurent Raymond, Yannik Thomas
We present a review of the work L. Raymond from 1995. The review aims at making this work more accessible and offers adaptations of some statements and proofs. In addition, this review forms an applicable framework for the complete solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in the work arXiv:2402.16703 by R. Band, S. Beckus and R. Loewy. A Sturmian Hamiltonian is a one-dimensional Schr"odinger operator whose potential is a Sturmian sequence multiplied by a coupling constant, $Vinmathbb{R}$. The spectrum of such an operator is commonly approximated by the spectra of designated periodic operators. If $V>4$, then the spectral bands of the periodic operators exhibit a particular combinatorial structure. This structure provides a formula for the integrated density of states. Employing this, it is shown that if $V>4$, then all the gaps, as predicted by the gap labelling theorem, are there.
我们对 L. Raymond 1995 年的著作进行了回顾。这篇评论旨在使这一工作更易于理解，并对一些陈述和证明进行了调整。此外，这篇综述还形成了一个适用的框架，用于解决 R. Band、S. Beckus 和 R. Loewy 在 arXiv:2402.16703 号著作中提出的斯图尔缪哈密顿的干十马尔蒂尼问题。斯图尔绵哈密顿是一个一维薛定谔算子，它的势是一个斯图尔绵序列乘以一个耦合常数$Vinmathbb{R}$。这种算子的谱通常用指定周期算子的谱来近似。如果 $V>4$，那么周期算子的谱带就会表现出一种特殊的组合结构。这种结构提供了一个积分态密度公式。利用这个公式，可以证明如果 $V>4$，那么间隙标签定理所预言的所有间隙都存在。
{"title":"A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels","authors":"Ram Band, Siegfried Beckus, Barak Biber, Laurent Raymond, Yannik Thomas","doi":"arxiv-2409.10920","DOIUrl":"https://doi.org/arxiv-2409.10920","url":null,"abstract":"We present a review of the work L. Raymond from 1995. The review aims at\u0000making this work more accessible and offers adaptations of some statements and\u0000proofs. In addition, this review forms an applicable framework for the complete\u0000solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in\u0000the work arXiv:2402.16703 by R. Band, S. Beckus and R. Loewy. A Sturmian\u0000Hamiltonian is a one-dimensional Schr\"odinger operator whose potential is a\u0000Sturmian sequence multiplied by a coupling constant, $Vinmathbb{R}$. The\u0000spectrum of such an operator is commonly approximated by the spectra of\u0000designated periodic operators. If $V>4$, then the spectral bands of the\u0000periodic operators exhibit a particular combinatorial structure. This structure\u0000provides a formula for the integrated density of states. Employing this, it is\u0000shown that if $V>4$, then all the gaps, as predicted by the gap labelling\u0000theorem, are there.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248708","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 explain in which sense Schr"odinger operators with complex potentials appear to violate locality (or Weyl's asymptotics), and we pose three open problems related to this phenomenon.
{"title":"Open problem: Violation of locality for Schrödinger operators with complex potentials","authors":"Jean-Claude Cuenin, Rupert L. Frank","doi":"arxiv-2409.11285","DOIUrl":"https://doi.org/arxiv-2409.11285","url":null,"abstract":"We explain in which sense Schr\"odinger operators with complex potentials\u0000appear to violate locality (or Weyl's asymptotics), and we pose three open\u0000problems related to this phenomenon.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248706","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 class of stochastic matrices that have a stochastic $c$-th root for infinitely many natural numbers $c$ is introduced and studied. Such matrices are called arbitrarily finely divisible, and generalise the class of infinitely divisible matrices. In particular, if $A$ is a transition matrix for a Markov process over some time period, then arbitrarily finely divisibility of $A$ is the necessary and sufficient condition for the existence of transition matrices corresponding to this Markov process over arbitrarily short periods. In this paper, we lay the foundation for research into arbitrarily finely divisible matrices and demonstrate the concepts using specific examples of $2 times 2$ matrices, $3 times 3$ circulant matrices, and rank-two matrices.
{"title":"Arbitrarily Finely Divisible Matrices","authors":"Priyanka Joshi, Helena Šmigoc","doi":"arxiv-2409.11125","DOIUrl":"https://doi.org/arxiv-2409.11125","url":null,"abstract":"The class of stochastic matrices that have a stochastic $c$-th root for\u0000infinitely many natural numbers $c$ is introduced and studied. Such matrices\u0000are called arbitrarily finely divisible, and generalise the class of infinitely\u0000divisible matrices. In particular, if $A$ is a transition matrix for a Markov\u0000process over some time period, then arbitrarily finely divisibility of $A$ is\u0000the necessary and sufficient condition for the existence of transition matrices\u0000corresponding to this Markov process over arbitrarily short periods. In this paper, we lay the foundation for research into arbitrarily finely\u0000divisible matrices and demonstrate the concepts using specific examples of $2\u0000times 2$ matrices, $3 times 3$ circulant matrices, and rank-two matrices.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248707","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}
Two graphs $G$ and $H$ are emph{cospectral} if the adjacency matrices share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature, e.g. the famous GM-switching method. In this paper, we shall construct cospectral graphs via regular rational orthogonal matrix $Q$ with level two and three. We provide two straightforward algorithms to characterize with adjacency matrix $A$ of graph $G$ such that $Q^TAQ$ is again a (0,1)-matrix, and introduce two new switching methods to construct families of cospectral graphs which generalized the GM-switching to some extent.
{"title":"Constructing cospectral graphs via regular rational orthogonal matrix with level two and three","authors":"Lihuan Mao, Fu Yan","doi":"arxiv-2409.09998","DOIUrl":"https://doi.org/arxiv-2409.09998","url":null,"abstract":"Two graphs $G$ and $H$ are emph{cospectral} if the adjacency matrices share\u0000the same spectrum. Constructing cospectral non-isomorphic graphs has been\u0000studied extensively for many years and various constructions are known in the\u0000literature, e.g. the famous GM-switching method. In this paper, we shall\u0000construct cospectral graphs via regular rational orthogonal matrix $Q$ with\u0000level two and three. We provide two straightforward algorithms to characterize\u0000with adjacency matrix $A$ of graph $G$ such that $Q^TAQ$ is again a\u0000(0,1)-matrix, and introduce two new switching methods to construct families of\u0000cospectral graphs which generalized the GM-switching to some extent.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248713","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 investigates uniqueness results for perturbed periodic Schr"odinger operators on $mathbb{Z}^d$. Specifically, we consider operators of the form $H = -Delta + V + v$, where $Delta$ is the discrete Laplacian, $V: mathbb{Z}^d rightarrow mathbb{R}$ is a periodic potential, and $v: mathbb{Z}^d rightarrow mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-Delta u + V u + v u = lambda u$, for $lambda in mathbb{C}$, admits only the trivial solution $u equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of $lambda$.
本文研究了$mathbb{Z}^d$上扰动周期薛定谔算子的唯一性结果。具体来说，我们考虑了$H = -Delta + V + v$形式的算子，其中$Delta$是离散拉普拉奇，$V：是周期势，$v:mathbb{Z}^d rightarrow mathbb{C}$代表衰变的杂质。我们建立了定量条件，在这些条件下，方程 $-Delta u + V u + vu = lambda u$，对于 $lambda in mathbb{C}$，只接受微不足道的解$u equiv 0$。其主要应用包括：对于杂质衰减速度快于任何指数函数的运算符，不存在内嵌特征值；以及确定特征函数的急剧衰减率。我们的发现为杂质提供了精确的衰变条件，并分析了 $lambda$ 的不同谱系，从而扩展了以前的工作。
{"title":"Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators","authors":"Wencai Liu, Rodrigo Matos, John N. Treuer","doi":"arxiv-2409.10387","DOIUrl":"https://doi.org/arxiv-2409.10387","url":null,"abstract":"This paper investigates uniqueness results for perturbed periodic\u0000Schr\"odinger operators on $mathbb{Z}^d$. Specifically, we consider operators\u0000of the form $H = -Delta + V + v$, where $Delta$ is the discrete Laplacian,\u0000$V: mathbb{Z}^d rightarrow mathbb{R}$ is a periodic potential, and $v:\u0000mathbb{Z}^d rightarrow mathbb{C}$ represents a decaying impurity. We\u0000establish quantitative conditions under which the equation $-Delta u + V u + v\u0000u = lambda u$, for $lambda in mathbb{C}$, admits only the trivial solution\u0000$u equiv 0$. Key applications include the absence of embedded eigenvalues for\u0000operators with impurities decaying faster than any exponential function and the\u0000determination of sharp decay rates for eigenfunctions. Our findings extend\u0000previous works by providing precise decay conditions for impurities and\u0000analyzing different spectral regimes of $lambda$.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248710","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}
Let $H = -d^2/dx^2 + q(x)$, $x in mathbb{R}$, where $q(x)$ is a periodic potential, and suppose that the spectrum $sigma(H)$ of $H$ is the positive semi-axis $[0, infty)$. In the case where $q(x)$ is real-valued (and locally square-integrable) a well-known result of G. Borg states that $q(x)$ must vanish almost everywhere. However, as it was first observed by M.G. Gasymov, there is an abundance of complex-valued potentials for which $sigma(H) = [0, infty)$. In this article we conjecture a characterization of all complex-valued potentials whose spectrum is $[0, infty)$. We also present an analog of Borg's result for complex potentials.
{"title":"Non-Self-Adjoint Hill Operators whose Spectrum is a Real Interval","authors":"Vassilis G. Papanicolaou","doi":"arxiv-2409.10266","DOIUrl":"https://doi.org/arxiv-2409.10266","url":null,"abstract":"Let $H = -d^2/dx^2 + q(x)$, $x in mathbb{R}$, where $q(x)$ is a periodic\u0000potential, and suppose that the spectrum $sigma(H)$ of $H$ is the positive\u0000semi-axis $[0, infty)$. In the case where $q(x)$ is real-valued (and locally\u0000square-integrable) a well-known result of G. Borg states that $q(x)$ must\u0000vanish almost everywhere. However, as it was first observed by M.G. Gasymov,\u0000there is an abundance of complex-valued potentials for which $sigma(H) = [0,\u0000infty)$. In this article we conjecture a characterization of all complex-valued\u0000potentials whose spectrum is $[0, infty)$. We also present an analog of Borg's\u0000result for complex potentials.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248712","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 Newtonian potential operator for the Helmholtz equation, which is represented by the volume integral with fundamental solution as kernel function, is of great importance for direct and inverse scattering of acoustic waves. In this paper, the eigensystem for the Newtonian potential operator is firstly shown to be equivalent to that for the Helmholtz equation with nonlocal boundary condition for a bounded and simply connected Lipschitz-regular domain. Then, we compute explicitly the eigenvalues and eigenfunctions of the Newtonian potential operator when it is defined in a 3-dimensional ball. Furthermore, the eigenvalues' asymptotic behavior is demonstrated. To illustrate the behavior of certain eigenfunctions, some numerical simulations are included.
{"title":"Characterization of the Eigenvalues and Eigenfunctions of the Helmholtz Newtonian operator N^k","authors":"Zhe Wang, Ahcene Ghandriche, Jijun Liu","doi":"arxiv-2409.09394","DOIUrl":"https://doi.org/arxiv-2409.09394","url":null,"abstract":"The Newtonian potential operator for the Helmholtz equation, which is\u0000represented by the volume integral with fundamental solution as kernel\u0000function, is of great importance for direct and inverse scattering of acoustic\u0000waves. In this paper, the eigensystem for the Newtonian potential operator is\u0000firstly shown to be equivalent to that for the Helmholtz equation with nonlocal\u0000boundary condition for a bounded and simply connected Lipschitz-regular domain.\u0000Then, we compute explicitly the eigenvalues and eigenfunctions of the Newtonian\u0000potential operator when it is defined in a 3-dimensional ball. Furthermore, the\u0000eigenvalues' asymptotic behavior is demonstrated. To illustrate the behavior of\u0000certain eigenfunctions, some numerical simulations are included.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248711","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 the first counter-example showing that the ground state energy of electrons in an external Coulomb potential is not always a convex function of the number of electrons. This property had been conjectured to hold for decades and it plays an important role in quantum chemistry. Our counter-example involves an external potential generated by six nuclei of small fractional charges, placed far away from each other. The ground state energy of 3 electrons is proved to be higher than the average of the energies for 2 and 4 electrons. In addition, we show that the nuclei can bind 2 or 4 electrons, but not 3. Although the conjecture remains open for real nuclei (of integer charges), our work sets some doubt on the validity of the energy convexity for general atoms and molecules.
{"title":"The ground state energy is not always convex in the number of electrons","authors":"Simone Di Marino, Mathieu Lewin, Luca Nenna","doi":"arxiv-2409.08632","DOIUrl":"https://doi.org/arxiv-2409.08632","url":null,"abstract":"We provide the first counter-example showing that the ground state energy of\u0000electrons in an external Coulomb potential is not always a convex function of\u0000the number of electrons. This property had been conjectured to hold for decades\u0000and it plays an important role in quantum chemistry. Our counter-example\u0000involves an external potential generated by six nuclei of small fractional\u0000charges, placed far away from each other. The ground state energy of 3\u0000electrons is proved to be higher than the average of the energies for 2 and 4\u0000electrons. In addition, we show that the nuclei can bind 2 or 4 electrons, but\u0000not 3. Although the conjecture remains open for real nuclei (of integer\u0000charges), our work sets some doubt on the validity of the energy convexity for\u0000general atoms and molecules.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248716","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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