arXiv - MATH - Spectral Theory最新文献_第10页

A new approach to inverse Sturm-Liouville problems based on point interaction 基于点相互作用的 Sturm-Liouville 逆问题新方法

arXiv - MATH - Spectral Theory Pub Date : 2024-07-24 DOI: arxiv-2407.17223

Min Zhao, Jiangang Qi, Xiao Chen

引用次数: 0

Generalized Morse Functions, Excision and Higher Torsions 广义莫尔斯函数、切除和高阶扭转

arXiv - MATH - Spectral Theory Pub Date : 2024-07-24 DOI: arxiv-2407.17100

Martin Puchol, Junrong Yan

{"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}

引用次数: 0

Benjamini-Schramm and spectral convergence II. The non-homogeneous case Benjamini-Schramm 和频谱收敛 II.非均质情况

arXiv - MATH - Spectral Theory Pub Date : 2024-07-24 DOI: arxiv-2407.17264

Anton Deitmar

引用次数: 0

A short nonstandard proof of the Spectral Theorem for unbounded self-adjoint operators 无界自约算子谱定理的简短非标准证明

arXiv - MATH - Spectral Theory Pub Date : 2024-07-23 DOI: arxiv-2407.16136

Takashi Matsunaga

引用次数: 0

CR Paneitz operator on non-embeddable CR manifolds 不可嵌入CR流形上的CR帕尼茨算子

arXiv - MATH - Spectral Theory Pub Date : 2024-07-23 DOI: arxiv-2407.16185

Yuya Takeuchi

引用次数: 0

Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds 封闭黎曼流形上分数薛定谔算子的卡尔德龙问题

arXiv - MATH - Spectral Theory Pub Date : 2024-07-23 DOI: arxiv-2407.16866

Ali Feizmohammadi, Katya Krupchyk, Gunther Uhlmann

{"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}

引用次数: 0

Quantum Tunneling and the Aharonov-Bohm effect 量子隧道效应和阿哈诺夫-玻姆效应

arXiv - MATH - Spectral Theory Pub Date : 2024-07-23 DOI: arxiv-2407.16524

Bernard Helffer, Ayman Kachmar

引用次数: 0

Stability of quaternion matrix polynomials 四元矩阵多项式的稳定性

arXiv - MATH - Spectral Theory Pub Date : 2024-07-23 DOI: arxiv-2407.16603

Pallavi Basavaraju, Shrinath Hadimani, Sachindranath Jayaraman

{"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}

引用次数: 0

Minimizing Schrödinger eigenvalues for confining potentials 最小化约束势的薛定谔特征值

arXiv - MATH - Spectral Theory Pub Date : 2024-07-21 DOI: arxiv-2407.15103

Rupert L. Frank

引用次数: 0

Inverse spectral problem of Sturm-Liouville equation with many frozen arguments 具有多个冻结参数的 Sturm-Liouville 方程的逆谱问题

arXiv - MATH - Spectral Theory Pub Date : 2024-07-20 DOI: arxiv-2407.14889

Chung-Tsun Shieh, Tzong-Mo Tsai

引用次数: 0