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Quantitative spectral stability for compact operators 紧凑算子的定量谱稳定性
arXiv - MATH - Spectral Theory Pub Date : 2024-07-30 DOI: arxiv-2407.20809
Andrea Bisterzo, Giovanni Siclari
{"title":"Quantitative spectral stability for compact operators","authors":"Andrea Bisterzo, Giovanni Siclari","doi":"arxiv-2407.20809","DOIUrl":"https://doi.org/arxiv-2407.20809","url":null,"abstract":"This paper deals with quantitative spectral stability for compact operators\u0000acting on $L^2(X,m)$, where $(X,m)$ is a measure space. Under fairly general\u0000assumptions, we provide a characterization of the dominant term of the\u0000asymptotic expansion of the eigenvalue variation in this abstract setting. Many\u0000of the results about quantitative spectral stability available in the\u0000literature can be recovered by our analysis. Furthermore, we illustrate our\u0000result with several applications, e.g. quantitative spectral stability for a\u0000Robin to Neumann problem, conformal transformations of Riemann metrics,\u0000Dirichlet forms under the removal of sets of small capacity, and for families\u0000of pseudo-differentials operators.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869758","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 restricted discrete Fourier transform 受限离散傅立叶变换
arXiv - MATH - Spectral Theory Pub Date : 2024-07-29 DOI: arxiv-2407.20379
W. Riley Casper, Milen Yakimov
{"title":"The restricted discrete Fourier transform","authors":"W. Riley Casper, Milen Yakimov","doi":"arxiv-2407.20379","DOIUrl":"https://doi.org/arxiv-2407.20379","url":null,"abstract":"We investigate the restriction of the discrete Fourier transform $F_N :\u0000L^2(mathbb{Z}/N mathbb{Z}) to L^2(mathbb{Z}/N mathbb{Z})$ to the space\u0000$mathcal C_a$ of functions with support on the discrete interval $[-a,a]$,\u0000whose transforms are supported inside the same interval. A periodically\u0000tridiagonal matrix $J$ on $L^2(mathbb{Z}/N mathbb{Z})$ is constructed having\u0000the three properties that it commutes with $F_N$, has eigenspaces of dimensions\u00001 and 2 only, and the span of its eigenspaces of dimension 1 is precisely\u0000$mathcal C_a$. The simple eigenspaces of $J$ provide an orthonormal eigenbasis\u0000of the restriction of $F_N$ to $mathcal C_a$. The dimension 2 eigenspaces of\u0000$J$ have canonical basis elements supported on $[-a,a]$ and its complement.\u0000These bases give an interpolation formula for reconstructing $f(x)in\u0000L^2(mathbb{Z}/Nmathbb{Z})$ from the values of $f(x)$ and $widehat f(x)$ on\u0000$[-a,a]$, i.e., an explicit Fourier uniqueness pair interpolation formula. The\u0000coefficients of the interpolation formula are expressed in terms of theta\u0000functions. Lastly, we construct an explicit basis of $mathcal C_a$ having\u0000extremal support and leverage it to obtain explicit formulas for eigenfunctions\u0000of $F_N$ in $C_a$ when $dim mathcal C_a leq 4$.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"166 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869756","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
Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals 量化光谱尺寸的最佳算法及其在准晶体中的应用
arXiv - MATH - Spectral Theory Pub Date : 2024-07-29 DOI: arxiv-2407.20353
Matthew J. Colbrook, Mark Embree, Jake Fillman
{"title":"Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals","authors":"Matthew J. Colbrook, Mark Embree, Jake Fillman","doi":"arxiv-2407.20353","DOIUrl":"https://doi.org/arxiv-2407.20353","url":null,"abstract":"We introduce computational strategies for measuring the ``size'' of the\u0000spectrum of bounded self-adjoint operators using various metrics such as the\u0000Lebesgue measure, fractal dimensions, the number of connected components (or\u0000gaps), and other spectral characteristics. Our motivation comes from the study\u0000of almost-periodic operators, particularly those that arise as models of\u0000quasicrystals. Such operators are known for intricate hierarchical patterns and\u0000often display delicate spectral properties, such as Cantor spectra, which are\u0000significant in studying quantum mechanical systems and materials science. We\u0000propose a series of algorithms that compute these properties under different\u0000assumptions and explore their theoretical implications through the Solvability\u0000Complexity Index (SCI) hierarchy. This approach provides a rigorous framework\u0000for understanding the computational feasibility of these problems, proving\u0000algorithmic optimality, and enhancing the precision of spectral analysis in\u0000practical settings. For example, we show that our methods are optimal by\u0000proving certain lower bounds (impossibility results) for the class of\u0000limit-periodic Schr\"odinger operators. We demonstrate our methods through\u0000state-of-the-art computations for aperiodic systems in one and two dimensions,\u0000effectively capturing these complex spectral characteristics. The results\u0000contribute significantly to connecting theoretical and computational aspects of\u0000spectral theory, offering insights that bridge the gap between abstract\u0000mathematical concepts and their practical applications in physical sciences and\u0000engineering. Based on our work, we conclude with conjectures and open problems\u0000regarding the spectral properties of specific models.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"173 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869757","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 sharp quantitative nonlinear Poincaré inequality on convex domains 凸域上的尖锐定量非线性庞加莱不等式
arXiv - MATH - Spectral Theory Pub Date : 2024-07-29 DOI: arxiv-2407.20373
Vincenzo Amato, Dorin Bucur, Ilaria Fragalà
{"title":"A sharp quantitative nonlinear Poincaré inequality on convex domains","authors":"Vincenzo Amato, Dorin Bucur, Ilaria Fragalà","doi":"arxiv-2407.20373","DOIUrl":"https://doi.org/arxiv-2407.20373","url":null,"abstract":"For any $p in ( 1, +infty)$, we give a new inequality for the first\u0000nontrivial Neumann eigenvalue $mu _ p (Omega, varphi)$ of the $p$-Laplacian\u0000on a convex domain $Omega subset mathbb{R}^N$ with a power-concave weight\u0000$varphi$. Our result improves the classical estimate in terms of the diameter,\u0000first stated in a seminal paper by Payne and Weinberger: we add in the lower\u0000bound an extra term depending on the second largest John semi-axis of $Omega$\u0000(equivalent to a power of the width in the special case $N = 2$). The power\u0000exponent in the extra term is sharp, and the constant in front of it is\u0000explicitly tracked, thus enlightening the interplay between space dimension,\u0000nonlinearity and power-concavity. Moreover, we attack the stability question:\u0000we prove that, if $mu _ p (Omega, varphi)$ is close to the lower bound, then\u0000$Omega$ is close to a thin cylinder, and $varphi$ is close to a function\u0000which is constant along its axis. As intermediate results, we establish a sharp\u0000$L^ infty$ estimate for the associated eigenfunctions, and we determine the\u0000asymptotic behaviour of $mu _ p (Omega, varphi)$ for varying weights and\u0000domains, including the case of collapsing geometries.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"114 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869760","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 asymptotics of Robin eigenvalues in the Dirichlet limit 论迪里希特极限中罗宾特征值的渐近性
arXiv - MATH - Spectral Theory Pub Date : 2024-07-28 DOI: arxiv-2407.19505
Roberto Ognibene
{"title":"On asymptotics of Robin eigenvalues in the Dirichlet limit","authors":"Roberto Ognibene","doi":"arxiv-2407.19505","DOIUrl":"https://doi.org/arxiv-2407.19505","url":null,"abstract":"We investigate the asymptotic behavior of the eigenvalues of the Laplacian\u0000with homogeneous Robin boundary conditions, when the (positive) Robin parameter\u0000is diverging. In this framework, since the convergence of the Robin eigenvalues\u0000to the Dirichlet ones is known, we address the question of quantifying the rate\u0000of such convergence. More precisely, in this work we identify the proper\u0000geometric quantity representing (asymptotically) the first term in the\u0000expansion of the eigenvalue variation: it is a novel notion of torsional\u0000rigidity. Then, by performing a suitable asymptotic analysis of both such\u0000quantity and its minimizer, we prove the first-order expansion of any Robin\u0000eigenvalue, in the Dirichlet limit. Moreover, the convergence rate of the\u0000corresponding eigenfunctions is obtained as well. We remark that all our\u0000spectral estimates are explicit and sharp, and cover both the cases of\u0000convergence to simple and multiple Dirichlet eigenvalues.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869761","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
Dynamical localization for random scattering zippers 随机散射拉链的动态定位
arXiv - MATH - Spectral Theory Pub Date : 2024-07-27 DOI: arxiv-2407.19158
Amine Khouildi, Hakim Boumaza
{"title":"Dynamical localization for random scattering zippers","authors":"Amine Khouildi, Hakim Boumaza","doi":"arxiv-2407.19158","DOIUrl":"https://doi.org/arxiv-2407.19158","url":null,"abstract":"This article establishes a proof of dynamical localization for a random\u0000scattering zipper model. The scattering zipper operator is the product of two\u0000unitary by blocks operators, multiplicatively perturbed on the left and right\u0000by random unitary phases. One of the operator is shifted so that this\u0000configuration produces a random 5-diagonal unitary operator per blocks. To\u0000prove the dynamical localization for this operator, we use the method of\u0000fractional moments. We first prove the continuity and strict positivity of the\u0000Lyapunov exponents in an annulus around the unit circle, which leads to the\u0000exponential decay of a power of the norm of the products of transfer matrices.\u0000We then establish an explicit formulation of the coefficients of the finite\u0000resolvent from the coefficients of the transfer matrices using Schur's\u0000complement. From this we deduce, through two reduction results, the exponential\u0000decay of the resolvent, from which we get the dynamical localization after\u0000proving that it also implies the exponential decay of moments of order $2$ of\u0000the resolvent.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869763","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
Hearing the shape of a drum by knocking around 敲敲打打听鼓声
arXiv - MATH - Spectral Theory Pub Date : 2024-07-26 DOI: arxiv-2407.18797
Xing Wang, Emmett L. Wyman, Yakun Xi
{"title":"Hearing the shape of a drum by knocking around","authors":"Xing Wang, Emmett L. Wyman, Yakun Xi","doi":"arxiv-2407.18797","DOIUrl":"https://doi.org/arxiv-2407.18797","url":null,"abstract":"We study a variation of Kac's question, \"Can one hear the shape of a drum?\"\u0000if we allow ourselves access to some additional information. In particular, we\u0000allow ourselves to ``hear\" the local Weyl counting function at each point on\u0000the manifold and ask if this is enough to uniquely recover the Riemannian\u0000metric. This is physically equivalent to asking whether one can determine the\u0000shape of a drum if one is allowed to knock at any place on the drum. We show\u0000that the answer to this question is ``yes\" provided the Laplace-Beltrami\u0000spectrum of the drum is simple. We also provide a counterexample illustrating\u0000why this hypothesis is necessary.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869759","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 Point Charges Interacting with Quasi-classical Electromagnetic Fields 与准经典电磁场相互作用的量子点电荷
arXiv - MATH - Spectral Theory Pub Date : 2024-07-26 DOI: arxiv-2407.18600
S. Breteaux, M. Correggi, M. Falconi, J. Faupin
{"title":"Quantum Point Charges Interacting with Quasi-classical Electromagnetic Fields","authors":"S. Breteaux, M. Correggi, M. Falconi, J. Faupin","doi":"arxiv-2407.18600","DOIUrl":"https://doi.org/arxiv-2407.18600","url":null,"abstract":"We study effective models describing systems of quantum particles interacting\u0000with quantized (electromagnetic) fields in the quasi-classical regime, i.e.,\u0000when the field's state shows a large average number of excitations. Once the\u0000field's degrees of freedom are traced out on factorized states, the reduced\u0000dynamics of the particles' system is described by an effective Schr\"{o}dinger\u0000operator keeping track of the field's state. We prove that, under suitable\u0000assumptions on the latter, such effective models are well-posed even if the\u0000particles are point-like, that is no ultraviolet cut-off is imposed on the\u0000interaction with quantum fields.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"360 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869762","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
Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation 伪微分算子极谱值的荷尔德连续性、Gabor 框架边界和饱和度
arXiv - MATH - Spectral Theory Pub Date : 2024-07-25 DOI: arxiv-2407.18065
Karlheinz Gröchenig, José Luis Romero, Michael Speckbacher
{"title":"Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation","authors":"Karlheinz Gröchenig, José Luis Romero, Michael Speckbacher","doi":"arxiv-2407.18065","DOIUrl":"https://doi.org/arxiv-2407.18065","url":null,"abstract":"We build on our recent results on the Lipschitz dependence of the extreme\u0000spectral values of one-parameter families of pseudodifferential operators with\u0000symbols in a weighted Sj\"ostrand class. We prove that larger symbol classes\u0000lead to H\"older continuity with respect to the parameter. This result is then\u0000used to investigate the behavior of frame bounds of families of Gabor systems\u0000$mathcal{G}(g,alphaLambda)$ with respect to the parameter $alpha>0$, where\u0000$Lambda$ is a set of non-uniform, relatively separated time-frequency shifts,\u0000and $gin M^1_s(mathbb{R}^d)$, $0leq sleq 2$. In particular, we show that\u0000the frame bounds depend continuously on $alpha$ if $gin M^1(mathbb{R}^d)$,\u0000and are H\"older continuous if $gin M^1_s(mathbb{R}^d)$, $0<sleq 2$, with\u0000the H\"older exponent explicitly given.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770789","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
Direct resonance problem for Rayleigh seismic surface waves 雷利地震面波的直接共振问题
arXiv - MATH - Spectral Theory Pub Date : 2024-07-24 DOI: arxiv-2407.17580
Samuele Sottile
{"title":"Direct resonance problem for Rayleigh seismic surface waves","authors":"Samuele Sottile","doi":"arxiv-2407.17580","DOIUrl":"https://doi.org/arxiv-2407.17580","url":null,"abstract":"In this paper we study the direct resonance problem for Rayleigh seismic\u0000surface waves and obtain a constraint on the location of resonances and\u0000establish a forbidden domain as the main result. In order to obtain the main\u0000result we make a Pekeris-Markushevich transformation of the Rayleigh system\u0000with free surface boundary condition such that we get a matrix\u0000Schr\"odinger-type form of it. We obtain parity and analytical properties of\u0000its fundamental solutions, which are needed to prove the main theorem. We\u0000construct a function made up by Rayleigh determinants factors, which is proven\u0000to be entire, of exponential type and in the Cartwright class and leads to the\u0000constraint on the location of resonances.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770797","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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