We consider discrete Schr"odinger operators with periodic potentials on periodic graphs. Their spectra consist of a finite number of bands. By "rolling up" a periodic graph along some appropriate directions we obtain periodic graphs of smaller dimensions called subcovering graphs. For example, rolling up a planar hexagonal lattice along different directions will lead to nanotubes with various chiralities. We show that the subcovering graph is asymptotically isospectral to the original periodic graph as the length of the "chiral" (roll up) vectors tends to infinity and get asymptotics of the band edges of the Schr"odinger operator on the subcovering graph. We also obtain a criterion for the subcovering graph to be just isospectral to the original periodic graph. By isospectrality of periodic graphs we mean that the spectra of the Schr"odinger operators on the graphs consist of the same number of bands and the corresponding bands coincide as sets.
{"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}
We prove Weyl laws for Schr"odinger operators with critically singular potentials on compact manifolds with boundary. We also improve the Weyl remainder estimates under the condition that the set of all periodic geodesic billiards has measure 0. These extend the classical results by Seeley, Ivrii and Melrose. The proof uses the Gaussian heat kernel bounds for short times and a perturbation argument involving the wave equation.
{"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}
We present a novel theoretical framework connecting k-component edge connectivity with spectral graph theory and homology theory to pro vide new insights into the resilience of real-world networks. By extending classical edge connectivity to higher-dimensional simplicial complexes, we derive tight spectral-homological bounds on the minimum number of edges that must be removed to ensure that all remaining components in the graph have size less than k. These bounds relate the spectra of graph and simplicial Laplacians to topological invariants from homology, establishing a multi-dimensional measure of network robustness. Our framework improves the understanding of network resilience in critical systems such as the Western U.S. power grid and European rail network, and we extend our analysis to random graphs and expander graphs to demonstrate the broad applicability of the method. Keywords: k-component edge connectivity, spectral graph theory, homology, simplicial complexes, network resilience, Betti numbers, algebraic connectivity, random graphs, expander graphs, infrastructure systems
我们提出了一个新颖的理论框架，将 k 分量边缘连通性与谱图理论和同调理论联系起来，为现实世界网络的恢复能力提供了新的视角。通过将经典边连接性扩展到高维简单复数，我们推导出了为确保图中所有剩余分量的大小小于 k 而必须去除的最小边数的紧谱-同调约束。这些约束将图谱和简单拉普拉斯与同调的拓扑不变式联系起来，从而建立了网络鲁棒性的多维衡量标准。我们的框架提高了人们对美国西部电网和欧洲铁路网等关键系统中网络鲁棒性的理解，我们还将分析扩展到随机图和扩展图，以证明该方法的广泛适用性。关键词：K-连通性、谱图理论、同源性、简单复数、网络弹性、贝蒂数、代数连通性、随机图、扩展图、基础设施系统
{"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}
We study the topological properties of spaces of reflectionless canonical systems. In this analysis, a key role is played by a natural action of the group $operatorname{PSL}(2,mathbb R)$ on these spaces.
{"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}
Antonio Capella, Christof Melcher, Lauro Morales, Ramón G. Plaza
This paper studies moving 180-degree N'eel walls in ferromagnetic thin films under the reduced model for the in-plane magnetization proposed by Capella, Melcher and Otto [5], in the case when a sufficiently weak external magnetic field is applied. It is shown that the linearization around the moving N'eel wall's phase determines a spectral problem that is a relatively bounded perturbation of the linearization around the static N'eel wall, which is the solution when the external magnetic field is set to zero and which is spectrally stable. Uniform resolvent-type estimates for the linearized operator around the static wall are established in order to prove the spectral stability of the moving wall upon application of perturbation theory for linear operators. The spectral analysis is the basis to prove, in turn, both the decaying properties of the generated semigroup and the nonlinear stability of the moving N'eel wall under small perturbations, in the case of a sufficiently weak external magnetic field. The stability of the static N'eel wall, which was established in a companion paper [4], plays a key role to obtain the main result.
本文根据 Capella、Melcher 和 Otto [5]提出的面内磁化还原模型，研究了在施加足够弱的外部磁场时，铁磁薄膜中的 180 度移动 N'eel 墙。研究表明，围绕运动钕磁墙相位的线性化决定了一个谱问题，它是围绕静态钕磁墙线性化的相对有界扰动，而静态钕磁墙是外磁场设为零时的解，它在光谱上是稳定的。建立了静态壁周围线性化算子的均匀解析型估计，以便在应用线性算子的扰动理论时证明运动壁的谱稳定性。在谱分析的基础上，反过来证明了在足够弱的外部磁场情况下，所产生的半群的衰减特性和运动镍镉墙在小扰动下的非线性稳定性。静态鳗鱼壁的稳定性在另一篇论文[4]中已经建立，它对获得主要结果起着关键作用。
{"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}
We analyze the spectrum of the hexagonal lattice graph with a vertex coupling which manifestly violates the time reversal invariance and at high energies it asymptotically decouples edges at even degree vertices; a comparison is made to the case when such a decoupling occurs at odd degree vertices. We also show that the spectral character does not change if the equilateral elementary cell of the lattice is dilated to have three different edge lengths, except that flat bands are absent if those are incommensurate.
{"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}
We prove that on convex domains, first mixed Laplace eigenfunctions have no interior critical points if the Dirichlet region is connected and sufficiently small. We use this result to construct a new family of polygonal domains for which Rauch's hot spots conjecture holds and to prove a new general theorem regarding the hot spots conjecture.
{"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}
For the Laplacian in spherical and hyperbolic spaces, Robin eigenvalues in two dimensions and Dirichlet eigenvalues in higher dimensions are shown to satisfy scaling inequalities analogous to the standard scale invariance of the Euclidean Laplacian. These results extend work of Langford and Laugesen to Robin problems and to Dirichlet problems in higher dimensions. In addition, scaled Robin eigenvalues behave exotically as the domain expands to a 2-sphere, tending to the spectrum of an exterior Robin problem.
{"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}
In this work, we consider a general time-periodic linear transport equation with integral source term. We prove the existence of a Floquet principal eigenvalue, namely a real number such that the equation rescaled by this number admits nonnegative periodic solutions. We also prove the exponential attractiveness of these solutions. The method relies on general spectral results about positive operators.
{"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}
This paper deals with the Sturm-Liouville problem that feature distribution potential, polynomial dependence on the spectral parameter in the first boundary condition, and analytical dependence, in the second one. We study an inverse spectral problem that consists in the recovery of the potential and the polynomials from some part of the spectrum. We for the first time prove local solvability and stability for this type of inverse problems. Furthermore, the necessary and sufficient conditions on the given subspectrum for the uniqueness of solution are found, and a reconstruction procedure is developed. Our main results can be applied to a variety of partial inverse problems. This is illustrated by an example of the Hochstadt-Lieberman-type problem with polynomial dependence on the spectral parameter in the both boundary conditions.
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：