This paper deals with quantitative spectral stability for compact operators acting on $L^2(X,m)$, where $(X,m)$ is a measure space. Under fairly general assumptions, we provide a characterization of the dominant term of the asymptotic expansion of the eigenvalue variation in this abstract setting. Many of the results about quantitative spectral stability available in the literature can be recovered by our analysis. Furthermore, we illustrate our result with several applications, e.g. quantitative spectral stability for a Robin to Neumann problem, conformal transformations of Riemann metrics, Dirichlet forms under the removal of sets of small capacity, and for families of pseudo-differentials operators.
{"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}
We investigate the restriction of the discrete Fourier transform $F_N : L^2(mathbb{Z}/N mathbb{Z}) to L^2(mathbb{Z}/N mathbb{Z})$ to the space $mathcal C_a$ of functions with support on the discrete interval $[-a,a]$, whose transforms are supported inside the same interval. A periodically tridiagonal matrix $J$ on $L^2(mathbb{Z}/N mathbb{Z})$ is constructed having the three properties that it commutes with $F_N$, has eigenspaces of dimensions 1 and 2 only, and the span of its eigenspaces of dimension 1 is precisely $mathcal C_a$. The simple eigenspaces of $J$ provide an orthonormal eigenbasis of the restriction of $F_N$ to $mathcal C_a$. The dimension 2 eigenspaces of $J$ have canonical basis elements supported on $[-a,a]$ and its complement. These bases give an interpolation formula for reconstructing $f(x)in L^2(mathbb{Z}/Nmathbb{Z})$ from the values of $f(x)$ and $widehat f(x)$ on $[-a,a]$, i.e., an explicit Fourier uniqueness pair interpolation formula. The coefficients of the interpolation formula are expressed in terms of theta functions. Lastly, we construct an explicit basis of $mathcal C_a$ having extremal support and leverage it to obtain explicit formulas for eigenfunctions of $F_N$ in $C_a$ when $dim mathcal C_a leq 4$.
{"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}
We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and other spectral characteristics. Our motivation comes from the study of almost-periodic operators, particularly those that arise as models of quasicrystals. Such operators are known for intricate hierarchical patterns and often display delicate spectral properties, such as Cantor spectra, which are significant in studying quantum mechanical systems and materials science. We propose a series of algorithms that compute these properties under different assumptions and explore their theoretical implications through the Solvability Complexity Index (SCI) hierarchy. This approach provides a rigorous framework for understanding the computational feasibility of these problems, proving algorithmic optimality, and enhancing the precision of spectral analysis in practical settings. For example, we show that our methods are optimal by proving certain lower bounds (impossibility results) for the class of limit-periodic Schr"odinger operators. We demonstrate our methods through state-of-the-art computations for aperiodic systems in one and two dimensions, effectively capturing these complex spectral characteristics. The results contribute significantly to connecting theoretical and computational aspects of spectral theory, offering insights that bridge the gap between abstract mathematical concepts and their practical applications in physical sciences and engineering. Based on our work, we conclude with conjectures and open problems regarding the spectral properties of specific models.
{"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}
For any $p in ( 1, +infty)$, we give a new inequality for the first nontrivial Neumann eigenvalue $mu _ p (Omega, varphi)$ of the $p$-Laplacian on a convex domain $Omega subset mathbb{R}^N$ with a power-concave weight $varphi$. Our result improves the classical estimate in terms of the diameter, first stated in a seminal paper by Payne and Weinberger: we add in the lower bound an extra term depending on the second largest John semi-axis of $Omega$ (equivalent to a power of the width in the special case $N = 2$). The power exponent in the extra term is sharp, and the constant in front of it is explicitly tracked, thus enlightening the interplay between space dimension, nonlinearity and power-concavity. Moreover, we attack the stability question: we prove that, if $mu _ p (Omega, varphi)$ is close to the lower bound, then $Omega$ is close to a thin cylinder, and $varphi$ is close to a function which is constant along its axis. As intermediate results, we establish a sharp $L^ infty$ estimate for the associated eigenfunctions, and we determine the asymptotic behaviour of $mu _ p (Omega, varphi)$ for varying weights and domains, including the case of collapsing geometries.
对于任意 $p in ( 1, +infty)$，我们给出了一个新的不等式，即在具有幂凹权重$varphi$的凸域$Omega subset mathbb{R}^N$上，$p$-Laplacian 的第一个非难 Neumann 特征值$mu _ p (Omega, varphi)$。我们的结果改进了佩恩和温伯格（Payne and Weinberger）在一篇开创性论文中首次提出的以直径为单位的经典估计：我们在下界添加了一个额外项，它取决于 $Omega$ 的第二大约翰半轴（在特殊情况下，相当于 $N = 2$ 宽度的幂次）。额外项中的幂指数是尖锐的，其前面的常数被明确地跟踪，从而揭示了空间维度、非线性和幂凹性之间的相互作用。此外，我们还讨论了稳定性问题：我们证明，如果 $mu _ p (Omega, varphi)$ 接近下界，那么$Omega$ 接近一个薄圆柱体，而 $varphi$ 接近一个沿其轴线恒定的函数。作为中间结果，我们为相关的特征函数建立了一个尖锐的$L^ infty$估计，并确定了不同权重和域（包括塌缩几何的情况）下$mu _ p (Omega, varphi)$的渐近行为。
{"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}
We investigate the asymptotic behavior of the eigenvalues of the Laplacian with homogeneous Robin boundary conditions, when the (positive) Robin parameter is diverging. In this framework, since the convergence of the Robin eigenvalues to the Dirichlet ones is known, we address the question of quantifying the rate of such convergence. More precisely, in this work we identify the proper geometric quantity representing (asymptotically) the first term in the expansion of the eigenvalue variation: it is a novel notion of torsional rigidity. Then, by performing a suitable asymptotic analysis of both such quantity and its minimizer, we prove the first-order expansion of any Robin eigenvalue, in the Dirichlet limit. Moreover, the convergence rate of the corresponding eigenfunctions is obtained as well. We remark that all our spectral estimates are explicit and sharp, and cover both the cases of convergence to simple and multiple Dirichlet eigenvalues.
{"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}
This article establishes a proof of dynamical localization for a random scattering zipper model. The scattering zipper operator is the product of two unitary by blocks operators, multiplicatively perturbed on the left and right by random unitary phases. One of the operator is shifted so that this configuration produces a random 5-diagonal unitary operator per blocks. To prove the dynamical localization for this operator, we use the method of fractional moments. We first prove the continuity and strict positivity of the Lyapunov exponents in an annulus around the unit circle, which leads to the exponential decay of a power of the norm of the products of transfer matrices. We then establish an explicit formulation of the coefficients of the finite resolvent from the coefficients of the transfer matrices using Schur's complement. From this we deduce, through two reduction results, the exponential decay of the resolvent, from which we get the dynamical localization after proving that it also implies the exponential decay of moments of order $2$ of the resolvent.
{"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}
We study a variation of Kac's question, "Can one hear the shape of a drum?" if we allow ourselves access to some additional information. In particular, we allow ourselves to ``hear" the local Weyl counting function at each point on the manifold and ask if this is enough to uniquely recover the Riemannian metric. This is physically equivalent to asking whether one can determine the shape of a drum if one is allowed to knock at any place on the drum. We show that the answer to this question is ``yes" provided the Laplace-Beltrami spectrum of the drum is simple. We also provide a counterexample illustrating why this hypothesis is necessary.
{"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}
We study effective models describing systems of quantum particles interacting with quantized (electromagnetic) fields in the quasi-classical regime, i.e., when the field's state shows a large average number of excitations. Once the field's degrees of freedom are traced out on factorized states, the reduced dynamics of the particles' system is described by an effective Schr"{o}dinger operator keeping track of the field's state. We prove that, under suitable assumptions on the latter, such effective models are well-posed even if the particles are point-like, that is no ultraviolet cut-off is imposed on the interaction with quantum fields.
{"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}
Karlheinz Gröchenig, José Luis Romero, Michael Speckbacher
We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sj"ostrand class. We prove that larger symbol classes lead to H"older continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $mathcal{G}(g,alphaLambda)$ with respect to the parameter $alpha>0$, where $Lambda$ is a set of non-uniform, relatively separated time-frequency shifts, and $gin M^1_s(mathbb{R}^d)$, $0leq sleq 2$. In particular, we show that the frame bounds depend continuously on $alpha$ if $gin M^1(mathbb{R}^d)$, and are H"older continuous if $gin M^1_s(mathbb{R}^d)$, $0
{"title":"Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation","authors":"Karlheinz Gröchenig, José Luis Romero, Michael Speckbacher","doi":"arxiv-2407.18065","DOIUrl":"https://doi.org/arxiv-2407.18065","url":null,"abstract":"We build on our recent results on the Lipschitz dependence of the extreme\u0000spectral values of one-parameter families of pseudodifferential operators with\u0000symbols in a weighted Sj\"ostrand class. We prove that larger symbol classes\u0000lead to H\"older continuity with respect to the parameter. This result is then\u0000used to investigate the behavior of frame bounds of families of Gabor systems\u0000$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}
In this paper we study the direct resonance problem for Rayleigh seismic surface waves and obtain a constraint on the location of resonances and establish a forbidden domain as the main result. In order to obtain the main result we make a Pekeris-Markushevich transformation of the Rayleigh system with free surface boundary condition such that we get a matrix Schr"odinger-type form of it. We obtain parity and analytical properties of its fundamental solutions, which are needed to prove the main theorem. We construct a function made up by Rayleigh determinants factors, which is proven to be entire, of exponential type and in the Cartwright class and leads to the constraint on the location of resonances.
{"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}
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