In the present paper, motivated by point interaction, we propose a new and explicit approach to inverse Sturm-Liouville eigenvalue problems under Dirichlet boundary. More precisely, when a given Sturm-Liouville eigenvalue problem with the unknown integrable potential interacts with $delta$-function potentials, we obtain a family of perturbation problems, called point interaction models in quantum mechanics. Then, only depending on the first eigenvalues of these perturbed problems, we define and study the first eigenvalue function, by which the desired potential can be expressed explicitly and uniquely. As by-products, using the analytic function theoretic tools, we also generalize several fundamental theorems of classical Sturm-Liouville problems to measure differential equations.
{"title":"A new approach to inverse Sturm-Liouville problems based on point interaction","authors":"Min Zhao, Jiangang Qi, Xiao Chen","doi":"arxiv-2407.17223","DOIUrl":"https://doi.org/arxiv-2407.17223","url":null,"abstract":"In the present paper, motivated by point interaction, we propose a new and\u0000explicit approach to inverse Sturm-Liouville eigenvalue problems under\u0000Dirichlet boundary. More precisely, when a given Sturm-Liouville eigenvalue\u0000problem with the unknown integrable potential interacts with $delta$-function\u0000potentials, we obtain a family of perturbation problems, called point\u0000interaction models in quantum mechanics. Then, only depending on the first\u0000eigenvalues of these perturbed problems, we define and study the first\u0000eigenvalue function, by which the desired potential can be expressed explicitly\u0000and uniquely. As by-products, using the analytic function theoretic tools, we\u0000also generalize several fundamental theorems of classical Sturm-Liouville\u0000problems to measure differential equations.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770796","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}
Comparing invariants from both topological and geometric perspectives is a key focus in index theorem. This paper compares higher analytic and topological torsions and establishes a version of the higher Cheeger-M"uller/Bismut-Zhang theorem. In fact, Bismut-Goette achieved this comparison assuming the existence of fiberwise Morse functions satisfying the fiberwise Thom-Smale transversality condition (TS condition). To fully generalize the theorem, we should remove this assumption. Notably, unlike fiberwise Morse functions, fiberwise generalized Morse functions (GMFs) always exist, we extend Bismut-Goette's setup by considering a fibration $ M to S $ with a unitarily flat complex bundle $ F to M $ and a fiberwise GMF $ f $, while retaining the TS condition. Compared to Bismut-Goette's work, handling birth-death points for a generalized Morse function poses a key difficulty. To address this, first, by the work of the author M.P., joint with Zhang and Zhu, we focus on a relative version of the theorem. Here, analytic and topological torsions are normalized by subtracting their corresponding torsions for trivial bundles. Next, using new techniques from by the author J.Y., we excise a small neighborhood around the locus where $f$ has birth-death points. This reduces the problem to Bismut-Goette's settings (or its version with boundaries) via a Witten-type deformation. However, new difficulties arise from very singular critical points during this deformation.To address these, we extend methods from Bismut-Lebeau, using Agmon estimates for noncompact manifolds developed by Dai and J.Y.
从拓扑和几何角度比较不变式是索引定理的一个重点。本文比较了高等解析翘曲和拓扑翘曲，并建立了高等切格-穆勒/比斯穆特-张定理的一个版本。事实上，俾斯麦-高特是在假定存在满足纤维性 Thom-Smale 横向条件（TS 条件）的纤维性莫尔斯函数的情况下实现这一比较的。为了完全推广该定理，我们应该取消这一假设。值得注意的是，与纤维莫尔斯函数不同，纤维广义莫尔斯函数（GMFs）总是存在的，我们在保留TS条件的前提下，通过考虑纤维$ M to S $与单位平复束$ F to M $和纤维广义GMF $ f $，扩展了比斯穆特-戈埃特的设置。与比斯穆特-戈埃特的工作相比，处理广义莫尔斯函数的生灭点是一个关键难题。为了解决这个问题，首先，通过作者M.P.与张和朱的联合工作，我们重点研究了该定理的相对版本。在这里，解析扭转和拓扑扭转是通过减去琐细束的相应扭转来归一化的。接下来，我们利用作者 J.Y. 的新技术，在$f$有出生-死亡点的位置周围切除一个小邻域。这就通过维滕类型变换将问题简化为俾斯麦-戈埃特（Bismut-Goette）的设置（或其有边界的版本）。为了解决这些问题，我们扩展了俾斯麦-勒博的方法，使用戴建华和 J.Y. 提出的非紧凑流形的阿格蒙估计。
{"title":"Generalized Morse Functions, Excision and Higher Torsions","authors":"Martin Puchol, Junrong Yan","doi":"arxiv-2407.17100","DOIUrl":"https://doi.org/arxiv-2407.17100","url":null,"abstract":"Comparing invariants from both topological and geometric perspectives is a\u0000key focus in index theorem. This paper compares higher analytic and topological\u0000torsions and establishes a version of the higher Cheeger-M\"uller/Bismut-Zhang\u0000theorem. In fact, Bismut-Goette achieved this comparison assuming the existence\u0000of fiberwise Morse functions satisfying the fiberwise Thom-Smale transversality\u0000condition (TS condition). To fully generalize the theorem, we should remove\u0000this assumption. Notably, unlike fiberwise Morse functions, fiberwise\u0000generalized Morse functions (GMFs) always exist, we extend Bismut-Goette's\u0000setup by considering a fibration $ M to S $ with a unitarily flat complex\u0000bundle $ F to M $ and a fiberwise GMF $ f $, while retaining the TS condition. Compared to Bismut-Goette's work, handling birth-death points for a\u0000generalized Morse function poses a key difficulty. To address this, first, by\u0000the work of the author M.P., joint with Zhang and Zhu, we focus on a relative\u0000version of the theorem. Here, analytic and topological torsions are normalized\u0000by subtracting their corresponding torsions for trivial bundles. Next, using\u0000new techniques from by the author J.Y., we excise a small neighborhood around\u0000the locus where $f$ has birth-death points. This reduces the problem to\u0000Bismut-Goette's settings (or its version with boundaries) via a Witten-type\u0000deformation. However, new difficulties arise from very singular critical points\u0000during this deformation.To address these, we extend methods from Bismut-Lebeau,\u0000using Agmon estimates for noncompact manifolds developed by Dai and J.Y.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770808","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 equivalence of spectral convergence and Benjamini-Schramm convergence is extended from homogeneous spaces to spaces which are compact modulo isometry group. The equivalence is proven under the condition of a uniform discreteness property. It is open, which implications hold without this condition.
{"title":"Benjamini-Schramm and spectral convergence II. The non-homogeneous case","authors":"Anton Deitmar","doi":"arxiv-2407.17264","DOIUrl":"https://doi.org/arxiv-2407.17264","url":null,"abstract":"The equivalence of spectral convergence and Benjamini-Schramm convergence is\u0000extended from homogeneous spaces to spaces which are compact modulo isometry\u0000group. The equivalence is proven under the condition of a uniform discreteness\u0000property. It is open, which implications hold without this condition.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770795","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}
By nonstandard analysis, a very short and elementary proof of the Spectral Theorem for unbounded self-adjoint operators is given.
通过非标准分析，给出了无界自约算子谱定理的简短而基本的证明。
{"title":"A short nonstandard proof of the Spectral Theorem for unbounded self-adjoint operators","authors":"Takashi Matsunaga","doi":"arxiv-2407.16136","DOIUrl":"https://doi.org/arxiv-2407.16136","url":null,"abstract":"By nonstandard analysis, a very short and elementary proof of the Spectral\u0000Theorem for unbounded self-adjoint operators is given.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770800","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 CR Paneitz operator is closely related to some important problems in CR geometry. In this paper, we consider this operator on a non-embeddable CR manifold. This operator is essentially self-adjoint and its spectrum is discrete except zero. Moreover, the eigenspace corresponding to each non-zero eigenvalue is a finite dimensional subspace of the space of smooth functions. Furthermore, we show that the CR Paneitz operator on the Rossi sphere, an example of non-embeddable CR manifolds, has infinitely many negative eigenvalues, which is significantly different from the embeddable case.
{"title":"CR Paneitz operator on non-embeddable CR manifolds","authors":"Yuya Takeuchi","doi":"arxiv-2407.16185","DOIUrl":"https://doi.org/arxiv-2407.16185","url":null,"abstract":"The CR Paneitz operator is closely related to some important problems in CR\u0000geometry. In this paper, we consider this operator on a non-embeddable CR\u0000manifold. This operator is essentially self-adjoint and its spectrum is\u0000discrete except zero. Moreover, the eigenspace corresponding to each non-zero\u0000eigenvalue is a finite dimensional subspace of the space of smooth functions.\u0000Furthermore, we show that the CR Paneitz operator on the Rossi sphere, an\u0000example of non-embeddable CR manifolds, has infinitely many negative\u0000eigenvalues, which is significantly different from the embeddable case.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770801","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}
Ali Feizmohammadi, Katya Krupchyk, Gunther Uhlmann
We study an analog of the anisotropic Calder'on problem for fractional Schr"odinger operators $(-Delta_g)^alpha + V$ with $alpha in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schr"odinger equation, given on an open nonempty a priori known subset of the manifold determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. Our method of proof is based on: (i) studying a new variant of the Gel'fand inverse spectral problem without the normalization assumption on the energy of eigenfunctions, and (ii) the discovery of an entanglement principle for nonlocal equations involving two or more compactly supported functions. Our solution to (i) makes connections to antipodal sets as well as local control for eigenfunctions and quantum chaos, while (ii) requires sharp interpolation results for holomorphic functions. We believe that both of these results can find applications in other areas of inverse problems.
{"title":"Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds","authors":"Ali Feizmohammadi, Katya Krupchyk, Gunther Uhlmann","doi":"arxiv-2407.16866","DOIUrl":"https://doi.org/arxiv-2407.16866","url":null,"abstract":"We study an analog of the anisotropic Calder'on problem for fractional\u0000Schr\"odinger operators $(-Delta_g)^alpha + V$ with $alpha in (0,1)$ on\u0000closed Riemannian manifolds of dimensions two and higher. We prove that the\u0000knowledge of a Cauchy data set of solutions of the fractional Schr\"odinger\u0000equation, given on an open nonempty a priori known subset of the manifold\u0000determines both the Riemannian manifold up to an isometry and the potential up\u0000to the corresponding gauge transformation, under certain geometric assumptions\u0000on the manifold as well as the observation set. Our method of proof is based\u0000on: (i) studying a new variant of the Gel'fand inverse spectral problem without\u0000the normalization assumption on the energy of eigenfunctions, and (ii) the\u0000discovery of an entanglement principle for nonlocal equations involving two or\u0000more compactly supported functions. Our solution to (i) makes connections to\u0000antipodal sets as well as local control for eigenfunctions and quantum chaos,\u0000while (ii) requires sharp interpolation results for holomorphic functions. We\u0000believe that both of these results can find applications in other areas of\u0000inverse problems.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"163 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770798","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 investigate a Hamiltonian with radial potential wells and an Aharonov-Bohm vector potential with two poles. Assuming that the potential wells are symmetric, we derive the semi-classical asymptotics of the splitting between the ground and second state energies. The flux effects due to the Aharonov-Bohm vector potential are of lower order compared to the contributions coming from the potential wells.
{"title":"Quantum Tunneling and the Aharonov-Bohm effect","authors":"Bernard Helffer, Ayman Kachmar","doi":"arxiv-2407.16524","DOIUrl":"https://doi.org/arxiv-2407.16524","url":null,"abstract":"We investigate a Hamiltonian with radial potential wells and an Aharonov-Bohm\u0000vector potential with two poles. Assuming that the potential wells are\u0000symmetric, we derive the semi-classical asymptotics of the splitting between\u0000the ground and second state energies. The flux effects due to the Aharonov-Bohm\u0000vector potential are of lower order compared to the contributions coming from\u0000the potential wells.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770807","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}
A right quaternion matrix polynomial is an expression of the form $P(lambda)= displaystyle sum_{i=0}^{m}A_i lambda^i$, where $A_i$'s are $n times n$ quaternion matrices with $A_m neq 0$. The aim of this manuscript is to determine the location of right eigenvalues of $P(lambda)$ relative to certain subsets of the set of quaternions. In particular, we extend the notion of (hyper)stability of complex matrix polynomials to quaternion matrix polynomials and obtain location of right eigenvalues of $P(lambda)$ using the following methods: $(1)$ we give a relation between (hyper)stability of a quaternion matrix polynomial and its complex adjoint matrix polynomial, $(2)$ we prove that $P(lambda)$ is stable with respect to an open (closed) ball in the set of quaternions, centered at a complex number if and only if it is stable with respect to its intersection with the set of complex numbers and $(3)$ as a consequence of $(1)$ and $(2)$, we prove that right eigenvalues of $P(lambda)$ lie between two concentric balls of specific radii in the set of quaternions centered at the origin. We identify classes of quaternion matrix polynomials for which stability and hyperstability are equivalent. We finally deduce hyperstability of certain univariate quaternion matrix polynomials via stability of certain multivariate quaternion matrix polynomials.
{"title":"Stability of quaternion matrix polynomials","authors":"Pallavi Basavaraju, Shrinath Hadimani, Sachindranath Jayaraman","doi":"arxiv-2407.16603","DOIUrl":"https://doi.org/arxiv-2407.16603","url":null,"abstract":"A right quaternion matrix polynomial is an expression of the form\u0000$P(lambda)= displaystyle sum_{i=0}^{m}A_i lambda^i$, where $A_i$'s are $n\u0000times n$ quaternion matrices with $A_m neq 0$. The aim of this manuscript is\u0000to determine the location of right eigenvalues of $P(lambda)$ relative to\u0000certain subsets of the set of quaternions. In particular, we extend the notion\u0000of (hyper)stability of complex matrix polynomials to quaternion matrix\u0000polynomials and obtain location of right eigenvalues of $P(lambda)$ using the\u0000following methods: $(1)$ we give a relation between (hyper)stability of a\u0000quaternion matrix polynomial and its complex adjoint matrix polynomial, $(2)$\u0000we prove that $P(lambda)$ is stable with respect to an open (closed) ball in\u0000the set of quaternions, centered at a complex number if and only if it is\u0000stable with respect to its intersection with the set of complex numbers and\u0000$(3)$ as a consequence of $(1)$ and $(2)$, we prove that right eigenvalues of\u0000$P(lambda)$ lie between two concentric balls of specific radii in the set of\u0000quaternions centered at the origin. We identify classes of quaternion matrix\u0000polynomials for which stability and hyperstability are equivalent. We finally\u0000deduce hyperstability of certain univariate quaternion matrix polynomials via\u0000stability of certain multivariate quaternion matrix polynomials.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770799","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 problem of minimizing the lowest eigenvalue of the Schr"odinger operator $-Delta+V$ in $L^2(mathbb R^d)$ when the integral $int e^{-tV},dx$ is given for some $t>0$. We show that the eigenvalue is minimal for the harmonic oscillator and derive a quantitative version of the corresponding inequality.
{"title":"Minimizing Schrödinger eigenvalues for confining potentials","authors":"Rupert L. Frank","doi":"arxiv-2407.15103","DOIUrl":"https://doi.org/arxiv-2407.15103","url":null,"abstract":"We consider the problem of minimizing the lowest eigenvalue of the\u0000Schr\"odinger operator $-Delta+V$ in $L^2(mathbb R^d)$ when the integral\u0000$int e^{-tV},dx$ is given for some $t>0$. We show that the eigenvalue is\u0000minimal for the harmonic oscillator and derive a quantitative version of the\u0000corresponding inequality.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770804","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 research was devoted to investigate the inverse spectral problem of Sturm-Liouville operator with many frozen arguments. Under some assumptions, the authors obtained uniqueness theorems. At the end, a numerical simulation for the inverse problem was presented.
{"title":"Inverse spectral problem of Sturm-Liouville equation with many frozen arguments","authors":"Chung-Tsun Shieh, Tzong-Mo Tsai","doi":"arxiv-2407.14889","DOIUrl":"https://doi.org/arxiv-2407.14889","url":null,"abstract":"This research was devoted to investigate the inverse spectral problem of\u0000Sturm-Liouville operator with many frozen arguments. Under some assumptions,\u0000the authors obtained uniqueness theorems. At the end, a numerical simulation\u0000for the inverse problem was presented.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770802","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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