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Topological and dynamical aspects of some spectral invariants of contact manifolds with circle action 具有圆作用的接触流形的一些谱不变量的拓扑和动力学方面
arXiv - MATH - Spectral Theory Pub Date : 2024-09-18 DOI: arxiv-2409.11787
Michel RuminLMO
{"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}
引用次数: 0
Uniform resolvent estimates, smoothing effects and spectral stability for the Heisenberg sublaplacian 海森堡亚拉普拉奇的均匀分解估计、平滑效应和频谱稳定性
arXiv - MATH - Spectral Theory Pub Date : 2024-09-18 DOI: arxiv-2409.11943
Luca Fanelli, Haruya Mizutani, Luz Roncal, Nico Michele Schiavone
{"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}
引用次数: 0
A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels 对雷蒙德著作的评论具有大耦合常数的斯图尔缪哈密顿--周期近似和间隙标签
arXiv - MATH - Spectral Theory Pub Date : 2024-09-17 DOI: arxiv-2409.10920
Ram Band, Siegfried Beckus, Barak Biber, Laurent Raymond, Yannik Thomas
{"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}
引用次数: 0
Open problem: Violation of locality for Schrödinger operators with complex potentials 未决问题：具有复势的薛定谔算子的违反局域性问题
arXiv - MATH - Spectral Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11285
Jean-Claude Cuenin, Rupert L. Frank
{"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}
引用次数: 0
Arbitrarily Finely Divisible Matrices 任意精细可分矩阵
arXiv - MATH - Spectral Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11125
Priyanka Joshi, Helena Šmigoc
{"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}
引用次数: 0
Constructing cospectral graphs via regular rational orthogonal matrix with level two and three 通过二级和三级规则有理正交矩阵构建余谱图
arXiv - MATH - Spectral Theory Pub Date : 2024-09-16 DOI: arxiv-2409.09998
Lihuan Mao, Fu Yan
{"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}
引用次数: 0
Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators 扰动周期薛定谔算子特征函数的锐衰减率
arXiv - MATH - Spectral Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10387
Wencai Liu, Rodrigo Matos, John N. Treuer
{"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}
引用次数: 0
Non-Self-Adjoint Hill Operators whose Spectrum is a Real Interval 频谱为实数区间的非自交希尔算子
arXiv - MATH - Spectral Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10266
Vassilis G. Papanicolaou
{"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}
引用次数: 0
Characterization of the Eigenvalues and Eigenfunctions of the Helmholtz Newtonian operator N^k 亥姆霍兹牛顿算子 N^k 的特征值和特征函数的表征
arXiv - MATH - Spectral Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09394
Zhe Wang, Ahcene Ghandriche, Jijun Liu
{"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}
引用次数: 0
The ground state energy is not always convex in the number of electrons 基态能量并不总是与电子数呈凸性关系
arXiv - MATH - Spectral Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08632
Simone Di Marino, Mathieu Lewin, Luca Nenna
{"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}
引用次数: 0
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