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Boundary spectral estimates for semiclassical Gevrey operators 半经典 Gevrey 算子的边界谱估计
arXiv - MATH - Spectral Theory Pub Date : 2024-08-17 DOI: arxiv-2408.09098
Haoren Xiong
{"title":"Boundary spectral estimates for semiclassical Gevrey operators","authors":"Haoren Xiong","doi":"arxiv-2408.09098","DOIUrl":"https://doi.org/arxiv-2408.09098","url":null,"abstract":"We obtain the spectral and resolvent estimates for semiclassical\u0000pseudodifferential operators with symbol of Gevrey-$s$ regularity, near the\u0000boundary of the range of the principal symbol. We prove that the boundary\u0000spectrum free region is of size ${mathcal O}(h^{1-frac{1}{s}})$ where the\u0000resolvent is at most fractional exponentially large in $h$, as the\u0000semiclassical parameter $hto 0^+$. This is a natural Gevrey analogue of a\u0000result by N. Dencker, J. Sj{\"o}strand, and M. Zworski in the $C^{infty}$ and\u0000analytic cases.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179321","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 Approximation for substitution systems 替代系统的频谱近似法
arXiv - MATH - Spectral Theory Pub Date : 2024-08-17 DOI: arxiv-2408.09282
Ram Band, Siegfried Beckus, Felix Pogorzelski, Lior Tenenbaum
{"title":"Spectral Approximation for substitution systems","authors":"Ram Band, Siegfried Beckus, Felix Pogorzelski, Lior Tenenbaum","doi":"arxiv-2408.09282","DOIUrl":"https://doi.org/arxiv-2408.09282","url":null,"abstract":"We study periodic approximations of aperiodic Schr\"odinger operators on\u0000lattices in Lie groups with dilation structure. The potentials arise through\u0000symbolic substitution systems that have been recently introduced in this\u0000setting. We characterize convergence of spectra of associated Schr\"odinger\u0000operators in the Hausdorff distance via properties of finite graphs. As a\u0000consequence, new examples of periodic approximations are obtained. We further\u0000prove that there are substitution systems that do not admit periodic\u0000approximations in higher dimensions, in contrast to the one-dimensional case.\u0000On the other hand, if the spectra converge, then we show that the rate of\u0000convergence is necessarily exponentially fast. These results are new even for\u0000substitutions over $mathbb{Z}^d$.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179312","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
Accelerating Spectral Clustering on Quantum and Analog Platforms 在量子和模拟平台上加速频谱聚类
arXiv - MATH - Spectral Theory Pub Date : 2024-08-16 DOI: arxiv-2408.08486
Xingzi Xu, Tuhin Sahai
{"title":"Accelerating Spectral Clustering on Quantum and Analog Platforms","authors":"Xingzi Xu, Tuhin Sahai","doi":"arxiv-2408.08486","DOIUrl":"https://doi.org/arxiv-2408.08486","url":null,"abstract":"We introduce a novel hybrid quantum-analog algorithm to perform graph\u0000clustering that exploits connections between the evolution of dynamical systems\u0000on graphs and the underlying graph spectra. This approach constitutes a new\u0000class of algorithms that combine emerging quantum and analog platforms to\u0000accelerate computations. Our hybrid algorithm is equivalent to spectral\u0000clustering and has a computational complexity of $O(N)$, where $N$ is the\u0000number of nodes in the graph, compared to $O(N^3)$ scaling on classical\u0000computing platforms. The proposed method employs the dynamic mode decomposition\u0000(DMD) framework on data generated by Schr\"{o}dinger dynamics embedded into the\u0000manifold generated by the graph Laplacian. We prove and demonstrate that one\u0000can extract the eigenvalues and scaled eigenvectors of the normalized graph\u0000Laplacian from quantum evolution on the graph by using DMD computations.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179317","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
Inequalities for eigenvalues of Schrödinger operators with mixed boundary conditions 具有混合边界条件的薛定谔算子特征值的不等式
arXiv - MATH - Spectral Theory Pub Date : 2024-08-16 DOI: arxiv-2409.00019
Nausica Aldeghi
{"title":"Inequalities for eigenvalues of Schrödinger operators with mixed boundary conditions","authors":"Nausica Aldeghi","doi":"arxiv-2409.00019","DOIUrl":"https://doi.org/arxiv-2409.00019","url":null,"abstract":"We consider the eigenvalue problem for the Schr\"odinger operator on bounded,\u0000convex domains with mixed boundary conditions, where a Dirichlet boundary\u0000condition is imposed on a part of the boundary and a Neumann boundary condition\u0000on its complement. We prove inequalities between the lowest eigenvalues\u0000corresponding to two different choices of such boundary conditions on both\u0000planar and higher-dimensional domains. We also prove an inequality between\u0000higher order mixed eigenvalues and pure Dirichlet eigenvalues on\u0000multidimensional polyhedral domains.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179319","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 ergodic theorems for semigroup representations 半群表示的均匀遍历定理
arXiv - MATH - Spectral Theory Pub Date : 2024-08-16 DOI: arxiv-2408.08961
Jochen Glück, Patrick Hermle, Henrik Kreidler
{"title":"Uniform ergodic theorems for semigroup representations","authors":"Jochen Glück, Patrick Hermle, Henrik Kreidler","doi":"arxiv-2408.08961","DOIUrl":"https://doi.org/arxiv-2408.08961","url":null,"abstract":"We consider a bounded representation $T$ of a commutative semigroup $S$ on a\u0000Banach space and analyse the relation between three concepts: (i) properties of\u0000the unitary spectrum of $T$, which is defined in terms of semigroup characters\u0000on $S$; (ii) uniform mean ergodic properties of $T$; and (iii)\u0000quasi-compactness of $T$. We use our results to generalize the celebrated Niiro-Sawashima theorem to\u0000semigroup representations and, as a consequence, obtain the following: if a\u0000positive and bounded semigroup representation on a Banach lattice is uniformly\u0000mean ergodic and has finite-dimensional fixed space, then it is quasi-compact.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179316","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 bulk-edge correspondence for curved interfaces 曲线界面的体边对应关系
arXiv - MATH - Spectral Theory Pub Date : 2024-08-15 DOI: arxiv-2408.07950
Alexis Drouot, Xiaowen Zhu
{"title":"The bulk-edge correspondence for curved interfaces","authors":"Alexis Drouot, Xiaowen Zhu","doi":"arxiv-2408.07950","DOIUrl":"https://doi.org/arxiv-2408.07950","url":null,"abstract":"The bulk-edge correspondence is a condensed matter theorem that relates the\u0000conductance of a Hall insulator in a half-plane to that of its (straight)\u0000boundary. In this work, we extend this result to domains with curved\u0000boundaries. Under mild geometric assumptions, we prove that the edge conductance of a\u0000topological insulator sample is an integer multiple of its Hall conductance.\u0000This integer counts the algebraic number of times that the interface (suitably\u0000oriented) enters the measurement set. This result provides a rigorous proof of\u0000a well-known experimental observation: arbitrarily truncated topological\u0000insulators support edge currents, regardless of the shape of their boundary.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179318","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
On the First Eigenvalue of the $p$-Laplace Operator with Robin Boundary Conditions in the Complement of a Compact Set 论紧凑集补集中具有罗宾边界条件的 $p$ 拉普拉斯算子的第一个特征值
arXiv - MATH - Spectral Theory Pub Date : 2024-08-12 DOI: arxiv-2408.06236
Lukas Bundrock, Tiziana Giorgi, Robert Smits
{"title":"On the First Eigenvalue of the $p$-Laplace Operator with Robin Boundary Conditions in the Complement of a Compact Set","authors":"Lukas Bundrock, Tiziana Giorgi, Robert Smits","doi":"arxiv-2408.06236","DOIUrl":"https://doi.org/arxiv-2408.06236","url":null,"abstract":"We consider the first eigenvalue $lambda_1$ of the $p$-Laplace operator\u0000subject to Robin boundary conditions in the exterior of a compact set. We\u0000discuss the conditions for the existence of a variational $lambda_1$,\u0000depending on the boundary parameter, the space dimension, and $p$. Our analysis\u0000involves the first $p$-harmonic Steklov eigenvalue in exterior domains. We\u0000establish properties of $lambda_1$ for the exterior of a ball, including\u0000general inequalities, the asymptotic behavior as the boundary parameter\u0000approaches zero, and a monotonicity result with respect to a special type of\u0000domain inclusion. In two dimensions, we generalized to $pneq 2$ some known\u0000shape optimization results.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179327","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
Negative eigenvalue estimates for the 1D Schr{ö}dinger operator with measure-potential 具有量势的一维薛定谔算子的负特征值估计
arXiv - MATH - Spectral Theory Pub Date : 2024-08-12 DOI: arxiv-2408.05980
Robert Fulsche, Medet Nursultanov, Grigori Rozenblum
{"title":"Negative eigenvalue estimates for the 1D Schr{ö}dinger operator with measure-potential","authors":"Robert Fulsche, Medet Nursultanov, Grigori Rozenblum","doi":"arxiv-2408.05980","DOIUrl":"https://doi.org/arxiv-2408.05980","url":null,"abstract":"We investigate the negative part of the spectrum of the operator $-partial^2\u0000- mu$ on $L^2(mathbb R)$, where a locally finite Radon measure $mu geq 0$\u0000is serving as a potential. We obtain estimates for the eigenvalue counting\u0000function, for individual eigenvalues and estimates of the Lieb-Thirring type. A\u0000crucial tool for our estimates is Otelbaev's function, a certain average of the\u0000measure potential $mu$, which is used both in the proofs and the formulation\u0000of many of the results.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179320","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
Perturbative diagonalization and spectral gaps of quasiperiodic operators on $ell^2(Z^d)$ with monotone potentials 具有单调势的$ell^2(Z^d)$上准周期算子的惯性对角化和谱隙
arXiv - MATH - Spectral Theory Pub Date : 2024-08-10 DOI: arxiv-2408.05650
Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg
{"title":"Perturbative diagonalization and spectral gaps of quasiperiodic operators on $ell^2(Z^d)$ with monotone potentials","authors":"Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg","doi":"arxiv-2408.05650","DOIUrl":"https://doi.org/arxiv-2408.05650","url":null,"abstract":"We obtain a perturbative proof of localization for quasiperiodic operators on\u0000$ell^2(Z^d)$ with one-dimensional phase space and monotone sampling\u0000functions, in the regime of small hopping. The proof is based on an iterative\u0000scheme which can be considered as a local (in the energy and the phase) and\u0000convergent version of KAM-type diagonalization, whose result is a covariant\u0000family of uniformly localized eigenvalues and eigenvectors. We also proof that\u0000the spectra of such operators contain infinitely many gaps.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179322","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
Transfer and entanglement stability of property ($UW${normalsizeit{E}}) 属性的转移和纠缠稳定性 ($UW${normalsizeit{E}})
arXiv - MATH - Spectral Theory Pub Date : 2024-08-10 DOI: arxiv-2408.05433
Sinan Qiu, Lining Jiang
{"title":"Transfer and entanglement stability of property ($UW${normalsizeit{E}})","authors":"Sinan Qiu, Lining Jiang","doi":"arxiv-2408.05433","DOIUrl":"https://doi.org/arxiv-2408.05433","url":null,"abstract":"An operator $Tin B(H)$ is said to satisfy property ($UW${scriptsize\u0000it{E}}) if the complement in the approximate point spectrum of the essential\u0000approximate point spectrum coincides with the isolated eigenvalues of the\u0000spectrum. Via the CI spectrum induced by consistent invertibility property of\u0000operators, we explore property ($UW${scriptsize it{E}}) for $T$ and $T^ast$\u0000simultaneously. Furthermore, the transfer of property ($UW${scriptsize\u0000it{E}}) from $T$ to $f(T)$ and $f(T^{ast})$ is obtained, where $f$ is a\u0000function which is analytic in a neighborhood of the spectrum of $T$. At last,\u0000with the help of the so-called $(A,B)$ entanglement stable spectra, the\u0000entanglement stability of property ($UW${scriptsize it{E}}) for $2times 2$\u0000upper triangular operator matrices is investigated.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179328","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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