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Spectrum of Schrödinger operators on subcovering graphs 子覆盖图上的薛定谔算子谱
arXiv - MATH - Spectral Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05830
Natalia Saburova
{"title":"Spectrum of Schrödinger operators on subcovering graphs","authors":"Natalia Saburova","doi":"arxiv-2409.05830","DOIUrl":"https://doi.org/arxiv-2409.05830","url":null,"abstract":"We consider discrete Schr\"odinger operators with periodic potentials on\u0000periodic graphs. Their spectra consist of a finite number of bands. By \"rolling\u0000up\" a periodic graph along some appropriate directions we obtain periodic\u0000graphs of smaller dimensions called subcovering graphs. For example, rolling up\u0000a planar hexagonal lattice along different directions will lead to nanotubes\u0000with various chiralities. We show that the subcovering graph is asymptotically\u0000isospectral to the original periodic graph as the length of the \"chiral\" (roll\u0000up) vectors tends to infinity and get asymptotics of the band edges of the\u0000Schr\"odinger operator on the subcovering graph. We also obtain a criterion for\u0000the subcovering graph to be just isospectral to the original periodic graph. By\u0000isospectrality of periodic graphs we mean that the spectra of the Schr\"odinger\u0000operators on the graphs consist of the same number of bands and the\u0000corresponding bands coincide as sets.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179340","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
Weyl laws for Schrödinger operators on compact manifolds with boundary 有边界紧凑流形上薛定谔算子的韦尔定律
arXiv - MATH - Spectral Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05252
Xiaoqi Huang, Xing Wang, Cheng Zhang
{"title":"Weyl laws for Schrödinger operators on compact manifolds with boundary","authors":"Xiaoqi Huang, Xing Wang, Cheng Zhang","doi":"arxiv-2409.05252","DOIUrl":"https://doi.org/arxiv-2409.05252","url":null,"abstract":"We prove Weyl laws for Schr\"odinger operators with critically singular\u0000potentials on compact manifolds with boundary. We also improve the Weyl\u0000remainder estimates under the condition that the set of all periodic geodesic\u0000billiards has measure 0. These extend the classical results by Seeley, Ivrii\u0000and Melrose. The proof uses the Gaussian heat kernel bounds for short times and\u0000a perturbation argument involving the wave equation.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179344","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
Spectral and Homological Bounds on k-Component Edge Connectivity k 分量边缘连通性的谱学和同调约束
arXiv - MATH - Spectral Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05725
Joshua Steier
{"title":"Spectral and Homological Bounds on k-Component Edge Connectivity","authors":"Joshua Steier","doi":"arxiv-2409.05725","DOIUrl":"https://doi.org/arxiv-2409.05725","url":null,"abstract":"We present a novel theoretical framework connecting k-component edge\u0000connectivity with spectral graph theory and homology theory to pro vide new\u0000insights into the resilience of real-world networks. By extending classical\u0000edge connectivity to higher-dimensional simplicial complexes, we derive tight\u0000spectral-homological bounds on the minimum number of edges that must be removed\u0000to ensure that all remaining components in the graph have size less than k.\u0000These bounds relate the spectra of graph and simplicial Laplacians to\u0000topological invariants from homology, establishing a multi-dimensional measure\u0000of network robustness. Our framework improves the understanding of network\u0000resilience in critical systems such as the Western U.S. power grid and European\u0000rail network, and we extend our analysis to random graphs and expander graphs\u0000to demonstrate the broad applicability of the method. Keywords: k-component\u0000edge connectivity, spectral graph theory, homology, simplicial complexes,\u0000network resilience, Betti numbers, algebraic connectivity, random graphs,\u0000expander graphs, infrastructure systems","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179342","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
Topological properties of reflectionless canonical systems 无反射典型系统的拓扑特性
arXiv - MATH - Spectral Theory Pub Date : 2024-09-07 DOI: arxiv-2409.04862
Max Forester, Christian Remling
{"title":"Topological properties of reflectionless canonical systems","authors":"Max Forester, Christian Remling","doi":"arxiv-2409.04862","DOIUrl":"https://doi.org/arxiv-2409.04862","url":null,"abstract":"We study the topological properties of spaces of reflectionless canonical\u0000systems. In this analysis, a key role is played by a natural action of the\u0000group $operatorname{PSL}(2,mathbb R)$ on these spaces.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179341","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
Stability of moving Néel walls in ferromagnetic thin films 铁磁薄膜中移动奈尔壁的稳定性
arXiv - MATH - Spectral Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04023
Antonio Capella, Christof Melcher, Lauro Morales, Ramón G. Plaza
{"title":"Stability of moving Néel walls in ferromagnetic thin films","authors":"Antonio Capella, Christof Melcher, Lauro Morales, Ramón G. Plaza","doi":"arxiv-2409.04023","DOIUrl":"https://doi.org/arxiv-2409.04023","url":null,"abstract":"This paper studies moving 180-degree N'eel walls in ferromagnetic thin films\u0000under the reduced model for the in-plane magnetization proposed by Capella,\u0000Melcher and Otto [5], in the case when a sufficiently weak external magnetic\u0000field is applied. It is shown that the linearization around the moving N'eel\u0000wall's phase determines a spectral problem that is a relatively bounded\u0000perturbation of the linearization around the static N'eel wall, which is the\u0000solution when the external magnetic field is set to zero and which is\u0000spectrally stable. Uniform resolvent-type estimates for the linearized operator\u0000around the static wall are established in order to prove the spectral stability\u0000of the moving wall upon application of perturbation theory for linear\u0000operators. The spectral analysis is the basis to prove, in turn, both the\u0000decaying properties of the generated semigroup and the nonlinear stability of\u0000the moving N'eel wall under small perturbations, in the case of a sufficiently\u0000weak external magnetic field. The stability of the static N'eel wall, which\u0000was established in a companion paper [4], plays a key role to obtain the main\u0000result.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179345","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
Spectral properties of hexagonal lattices with the -R coupling 具有 -R 耦合的六方晶格的光谱特性
arXiv - MATH - Spectral Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03538
Pavel Exner, Jan Pekař
{"title":"Spectral properties of hexagonal lattices with the -R coupling","authors":"Pavel Exner, Jan Pekař","doi":"arxiv-2409.03538","DOIUrl":"https://doi.org/arxiv-2409.03538","url":null,"abstract":"We analyze the spectrum of the hexagonal lattice graph with a vertex coupling\u0000which manifestly violates the time reversal invariance and at high energies it\u0000asymptotically decouples edges at even degree vertices; a comparison is made to\u0000the case when such a decoupling occurs at odd degree vertices. We also show\u0000that the spectral character does not change if the equilateral elementary cell\u0000of the lattice is dilated to have three different edge lengths, except that\u0000flat bands are absent if those are incommensurate.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179348","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 hot spots theorem for the mixed eigenvalue problem with small Dirichet region 具有小 Dirichet 区域的混合特征值问题的热点定理
arXiv - MATH - Spectral Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03908
Lawford Hatcher
{"title":"A hot spots theorem for the mixed eigenvalue problem with small Dirichet region","authors":"Lawford Hatcher","doi":"arxiv-2409.03908","DOIUrl":"https://doi.org/arxiv-2409.03908","url":null,"abstract":"We prove that on convex domains, first mixed Laplace eigenfunctions have no\u0000interior critical points if the Dirichlet region is connected and sufficiently\u0000small. We use this result to construct a new family of polygonal domains for\u0000which Rauch's hot spots conjecture holds and to prove a new general theorem\u0000regarding the hot spots conjecture.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179346","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
Scaling inequalities and limits for Robin and Dirichlet eigenvalues 罗宾特征值和德里赫特特征值的比例不等式和极限
arXiv - MATH - Spectral Theory Pub Date : 2024-09-04 DOI: arxiv-2409.03050
Scott Harman
{"title":"Scaling inequalities and limits for Robin and Dirichlet eigenvalues","authors":"Scott Harman","doi":"arxiv-2409.03050","DOIUrl":"https://doi.org/arxiv-2409.03050","url":null,"abstract":"For the Laplacian in spherical and hyperbolic spaces, Robin eigenvalues in\u0000two dimensions and Dirichlet eigenvalues in higher dimensions are shown to\u0000satisfy scaling inequalities analogous to the standard scale invariance of the\u0000Euclidean Laplacian. These results extend work of Langford and Laugesen to\u0000Robin problems and to Dirichlet problems in higher dimensions. In addition,\u0000scaled Robin eigenvalues behave exotically as the domain expands to a 2-sphere,\u0000tending to the spectrum of an exterior Robin problem.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179347","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 principal eigenvalue problem for time-periodic nonlocal equations with drift 有漂移的时周期非局部方程的主特征值问题
arXiv - MATH - Spectral Theory Pub Date : 2024-09-03 DOI: arxiv-2409.01868
Bertrand Cloez, Adil El Abdouni, Pierre Gabriel
{"title":"The principal eigenvalue problem for time-periodic nonlocal equations with drift","authors":"Bertrand Cloez, Adil El Abdouni, Pierre Gabriel","doi":"arxiv-2409.01868","DOIUrl":"https://doi.org/arxiv-2409.01868","url":null,"abstract":"In this work, we consider a general time-periodic linear transport equation\u0000with integral source term. We prove the existence of a Floquet principal\u0000eigenvalue, namely a real number such that the equation rescaled by this number\u0000admits nonnegative periodic solutions. We also prove the exponential\u0000attractiveness of these solutions. The method relies on general spectral\u0000results about positive operators.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179294","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
Inverse Sturm-Liouville problem with singular potential and spectral parameter in the boundary conditions 具有奇异势和边界条件谱参数的反斯特姆-刘维尔问题
arXiv - MATH - Spectral Theory Pub Date : 2024-09-03 DOI: arxiv-2409.02254
E. E. Chitorkin, N. P. Bondarenko
{"title":"Inverse Sturm-Liouville problem with singular potential and spectral parameter in the boundary conditions","authors":"E. E. Chitorkin, N. P. Bondarenko","doi":"arxiv-2409.02254","DOIUrl":"https://doi.org/arxiv-2409.02254","url":null,"abstract":"This paper deals with the Sturm-Liouville problem that feature distribution\u0000potential, polynomial dependence on the spectral parameter in the first\u0000boundary condition, and analytical dependence, in the second one. We study an\u0000inverse spectral problem that consists in the recovery of the potential and the\u0000polynomials from some part of the spectrum. We for the first time prove local\u0000solvability and stability for this type of inverse problems. Furthermore, the\u0000necessary and sufficient conditions on the given subspectrum for the uniqueness\u0000of solution are found, and a reconstruction procedure is developed. Our main\u0000results can be applied to a variety of partial inverse problems. This is\u0000illustrated by an example of the Hochstadt-Lieberman-type problem with\u0000polynomial dependence on the spectral parameter in the both boundary\u0000conditions.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179293","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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