We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey-$s$ regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is of size ${mathcal O}(h^{1-frac{1}{s}})$ where the resolvent is at most fractional exponentially large in $h$, as the semiclassical parameter $hto 0^+$. This is a natural Gevrey analogue of a result by N. Dencker, J. Sj{"o}strand, and M. Zworski in the $C^{infty}$ and analytic cases.
我们得到了具有 Gevrey-$s$ 正则符号的半经典伪微分算子在主符号范围边界附近的谱和解析量估计。我们证明，无边界谱区域的大小为 ${mathcal O}(h^{1-frac{1}{s}})$，其中溶剂在 $h$ 中最多是分数指数大，因为这些半经典参数 $hto 0^+$。这是 N. Dencker、J. Sj{"o}strand 和 M. Zworski 在$C^{infty}$ 和解析情况下得出的结果的自然 Gevrey 类比。
{"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}
Ram Band, Siegfried Beckus, Felix Pogorzelski, Lior Tenenbaum
We study periodic approximations of aperiodic Schr"odinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schr"odinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over $mathbb{Z}^d$.
{"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}
We introduce a novel hybrid quantum-analog algorithm to perform graph clustering that exploits connections between the evolution of dynamical systems on graphs and the underlying graph spectra. This approach constitutes a new class of algorithms that combine emerging quantum and analog platforms to accelerate computations. Our hybrid algorithm is equivalent to spectral clustering and has a computational complexity of $O(N)$, where $N$ is the number of nodes in the graph, compared to $O(N^3)$ scaling on classical computing platforms. The proposed method employs the dynamic mode decomposition (DMD) framework on data generated by Schr"{o}dinger dynamics embedded into the manifold generated by the graph Laplacian. We prove and demonstrate that one can extract the eigenvalues and scaled eigenvectors of the normalized graph Laplacian from quantum evolution on the graph by using DMD computations.
{"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}
We consider the eigenvalue problem for the Schr"odinger operator on bounded, convex domains with mixed boundary conditions, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. We prove inequalities between the lowest eigenvalues corresponding to two different choices of such boundary conditions on both planar and higher-dimensional domains. We also prove an inequality between higher order mixed eigenvalues and pure Dirichlet eigenvalues on multidimensional polyhedral domains.
{"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}
We consider a bounded representation $T$ of a commutative semigroup $S$ on a Banach space and analyse the relation between three concepts: (i) properties of the unitary spectrum of $T$, which is defined in terms of semigroup characters on $S$; (ii) uniform mean ergodic properties of $T$; and (iii) quasi-compactness of $T$. We use our results to generalize the celebrated Niiro-Sawashima theorem to semigroup representations and, as a consequence, obtain the following: if a positive and bounded semigroup representation on a Banach lattice is uniformly mean ergodic and has finite-dimensional fixed space, then it is quasi-compact.
{"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}
The bulk-edge correspondence is a condensed matter theorem that relates the conductance of a Hall insulator in a half-plane to that of its (straight) boundary. In this work, we extend this result to domains with curved boundaries. Under mild geometric assumptions, we prove that the edge conductance of a topological insulator sample is an integer multiple of its Hall conductance. This integer counts the algebraic number of times that the interface (suitably oriented) enters the measurement set. This result provides a rigorous proof of a well-known experimental observation: arbitrarily truncated topological insulators support edge currents, regardless of the shape of their boundary.
{"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}
We consider the first eigenvalue $lambda_1$ of the $p$-Laplace operator subject to Robin boundary conditions in the exterior of a compact set. We discuss the conditions for the existence of a variational $lambda_1$, depending on the boundary parameter, the space dimension, and $p$. Our analysis involves the first $p$-harmonic Steklov eigenvalue in exterior domains. We establish properties of $lambda_1$ for the exterior of a ball, including general inequalities, the asymptotic behavior as the boundary parameter approaches zero, and a monotonicity result with respect to a special type of domain inclusion. In two dimensions, we generalized to $pneq 2$ some known shape optimization results.
{"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}
Robert Fulsche, Medet Nursultanov, Grigori Rozenblum
We investigate the negative part of the spectrum of the operator $-partial^2 - mu$ on $L^2(mathbb R)$, where a locally finite Radon measure $mu geq 0$ is serving as a potential. We obtain estimates for the eigenvalue counting function, for individual eigenvalues and estimates of the Lieb-Thirring type. A crucial tool for our estimates is Otelbaev's function, a certain average of the measure potential $mu$, which is used both in the proofs and the formulation of many of the results.
{"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}
Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg
We obtain a perturbative proof of localization for quasiperiodic operators on $ell^2(Z^d)$ with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also proof that the spectra of such operators contain infinitely many gaps.
我们得到了关于$ell^2(Z^d)$上具有一维相空间和单调采样函数的准周期算子在小跳变制度下的局部化的微扰证明。证明基于一个迭代方案，该方案可视为 KAM 型对角化的局部（能量和相位）和收敛版本，其结果是一个均匀局部化特征值和特征向量的协方差族。我们还证明了这类算子的谱包含无穷多个间隙。
{"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}
An operator $Tin B(H)$ is said to satisfy property ($UW${scriptsize it{E}}) if the complement in the approximate point spectrum of the essential approximate point spectrum coincides with the isolated eigenvalues of the spectrum. Via the CI spectrum induced by consistent invertibility property of operators, we explore property ($UW${scriptsize it{E}}) for $T$ and $T^ast$ simultaneously. Furthermore, the transfer of property ($UW${scriptsize it{E}}) from $T$ to $f(T)$ and $f(T^{ast})$ is obtained, where $f$ is a function which is analytic in a neighborhood of the spectrum of $T$. At last, with the help of the so-called $(A,B)$ entanglement stable spectra, the entanglement stability of property ($UW${scriptsize it{E}}) for $2times 2$ upper triangular operator matrices is investigated.
{"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}
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