In this paper we prove sharp Lieb-Thirring (LT) inequalities for the family of shifted Coulomb Hamiltonians. More precisely, we prove the classical LT inequalities with the semi-classical constant for this family of operators in any dimension $dgeqslant 3$ and any $gamma geqslant 1$. We also prove that the semi-classical constant is never optimal for the Cwikel-Lieb-Rozenblum (CLR) inequalities for this family of operators in any dimension. In this case, we characterize the optimal constant as the minimum of a finite set and provide an asymptotic expansion as the dimension grows. Using the same method to prove the CLR inequalities for Coulomb, we obtain more information about the conjectured optimal constant in the CLR inequality for arbitrary potentials.
{"title":"Lieb-Thirring inequalities for the shifted Coulomb Hamiltonian","authors":"Thiago Carvalho Corso, Timo Weidl, Zhuoyao Zeng","doi":"arxiv-2409.01291","DOIUrl":"https://doi.org/arxiv-2409.01291","url":null,"abstract":"In this paper we prove sharp Lieb-Thirring (LT) inequalities for the family\u0000of shifted Coulomb Hamiltonians. More precisely, we prove the classical LT\u0000inequalities with the semi-classical constant for this family of operators in\u0000any dimension $dgeqslant 3$ and any $gamma geqslant 1$. We also prove that\u0000the semi-classical constant is never optimal for the Cwikel-Lieb-Rozenblum\u0000(CLR) inequalities for this family of operators in any dimension. In this case,\u0000we characterize the optimal constant as the minimum of a finite set and provide\u0000an asymptotic expansion as the dimension grows. Using the same method to prove\u0000the CLR inequalities for Coulomb, we obtain more information about the\u0000conjectured optimal constant in the CLR inequality for arbitrary potentials.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179295","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 show that every ergodic dynamical system induces a system with pure Lebesgue spectrum of infinite multiplicity.
我们证明，每个遍历动力系统都会诱导出一个具有无限倍率纯勒贝格谱的系统。
{"title":"Inducing countable Lebesgue spectrum","authors":"Fatna Abdedou, Bassam Fayad, Jean-Paul Thouvenot","doi":"arxiv-2409.00396","DOIUrl":"https://doi.org/arxiv-2409.00396","url":null,"abstract":"We show that every ergodic dynamical system induces a system with pure\u0000Lebesgue spectrum of infinite multiplicity.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179296","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 presents significant advancements in tensor analysis and the study of random walks on manifolds. It introduces new tensor inequalities derived using the Mond-Pecaric method, which enriches the existing mathematical tools for tensor analysis. This method, developed by mathematicians Mond and Pecaric, is a powerful technique for establishing inequalities in linear operators and matrices, using functional analysis and operator theory principles. The paper also proposes novel lower and upper bounds for estimating column sums of transition matrices based on their spectral information, which is critical for understanding random walk behavior. Additionally, it derives bounds for the right tail of weighted tensor sums derived from random walks on manifolds, utilizing the spectrum of the Laplace-Beltrami operator over the underlying manifolds and new tensor inequalities to enhance the understanding of these complex mathematical structures.
{"title":"Tail Bounds for Functions of Weighted Tensor Sums Derived from Random Walks on Riemannian Manifolds","authors":"Shih-Yu Chang","doi":"arxiv-2409.00542","DOIUrl":"https://doi.org/arxiv-2409.00542","url":null,"abstract":"This paper presents significant advancements in tensor analysis and the study\u0000of random walks on manifolds. It introduces new tensor inequalities derived\u0000using the Mond-Pecaric method, which enriches the existing mathematical tools\u0000for tensor analysis. This method, developed by mathematicians Mond and Pecaric,\u0000is a powerful technique for establishing inequalities in linear operators and\u0000matrices, using functional analysis and operator theory principles. The paper\u0000also proposes novel lower and upper bounds for estimating column sums of\u0000transition matrices based on their spectral information, which is critical for\u0000understanding random walk behavior. Additionally, it derives bounds for the\u0000right tail of weighted tensor sums derived from random walks on manifolds,\u0000utilizing the spectrum of the Laplace-Beltrami operator over the underlying\u0000manifolds and new tensor inequalities to enhance the understanding of these\u0000complex mathematical structures.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179299","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}
Many first-passage processes in complex media and related diffusion-controlled reactions can be described by means of eigenfunctions of the mixed Steklov-Neumann problem. In this paper, we investigate this spectral problem in a common setting when a small target or escape window (with Steklov condition) is located on the reflecting boundary (with Neumann condition). We start by inspecting two basic settings: an arc-shaped target on the boundary of a disk and a spherical-cap-shaped target on the boundary of a ball. We construct the explicit kernel of an integral operator that determines the eigenvalues and eigenfunctions and deduce their asymptotic behavior in the small-target limit. By relating the limiting kernel to an appropriate Dirichlet-to-Neumann operator, we extend these asymptotic results to other bounded domains with smooth boundaries. A straightforward application to first-passage processes is presented; in particular, we revisit the small-target behavior of the mean first-reaction time on perfectly or partially reactive targets, as well as for more sophisticated surface reactions that extend the conventional narrow escape problem.
{"title":"Mixed Steklov-Neumann problem: asymptotic analysis and applications to diffusion-controlled reactions","authors":"Denis S. Grebenkov","doi":"arxiv-2409.00213","DOIUrl":"https://doi.org/arxiv-2409.00213","url":null,"abstract":"Many first-passage processes in complex media and related\u0000diffusion-controlled reactions can be described by means of eigenfunctions of\u0000the mixed Steklov-Neumann problem. In this paper, we investigate this spectral\u0000problem in a common setting when a small target or escape window (with Steklov\u0000condition) is located on the reflecting boundary (with Neumann condition). We\u0000start by inspecting two basic settings: an arc-shaped target on the boundary of\u0000a disk and a spherical-cap-shaped target on the boundary of a ball. We\u0000construct the explicit kernel of an integral operator that determines the\u0000eigenvalues and eigenfunctions and deduce their asymptotic behavior in the\u0000small-target limit. By relating the limiting kernel to an appropriate\u0000Dirichlet-to-Neumann operator, we extend these asymptotic results to other\u0000bounded domains with smooth boundaries. A straightforward application to\u0000first-passage processes is presented; in particular, we revisit the\u0000small-target behavior of the mean first-reaction time on perfectly or partially\u0000reactive targets, as well as for more sophisticated surface reactions that\u0000extend the conventional narrow escape problem.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179297","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 construct a unitary operator between Hilbert spaces of generalized eigenfunctions of Coulomb operators and the Laplace-Beltrami operator of hyperbolic space that intertwines their respective Poisson operators on $L^2(mathbb{S}^{d-1})$. The constructed operator generalizes Fock's unitary transformation, originally defined between the discrete spectra of the attractive Coulomb operator and the Laplace-Beltrami operator on the sphere, to the setting of continuous spectra. Among other connections, this map explains why the scattering matrices are the same in these two different settings, and it also provides an explicit formula for the Poisson operator of the Coulomb Hamiltonian.
{"title":"On the intertwining map between Coulomb and hyperbolic scattering","authors":"Nicholas Lohr","doi":"arxiv-2408.16248","DOIUrl":"https://doi.org/arxiv-2408.16248","url":null,"abstract":"We construct a unitary operator between Hilbert spaces of generalized\u0000eigenfunctions of Coulomb operators and the Laplace-Beltrami operator of\u0000hyperbolic space that intertwines their respective Poisson operators on\u0000$L^2(mathbb{S}^{d-1})$. The constructed operator generalizes Fock's unitary\u0000transformation, originally defined between the discrete spectra of the\u0000attractive Coulomb operator and the Laplace-Beltrami operator on the sphere, to\u0000the setting of continuous spectra. Among other connections, this map explains\u0000why the scattering matrices are the same in these two different settings, and\u0000it also provides an explicit formula for the Poisson operator of the Coulomb\u0000Hamiltonian.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179304","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 obtain the sharp arithmetic Gordon's theorem: that is, absence of eigenvalues on the set of energies with Lyapunov exponent bounded by the exponential rate of approximation of frequency by the rationals, for a large class of one-dimensional quasiperiodic Schr"odinger operators, with no (modulus of) continuity required. The class includes all unbounded monotone potentials with finite Lyapunov exponents and all potentials of bounded variation. The main tool is a new uniform upper bound on iterates of cocycles of bounded variation.
{"title":"Sharp arithmetic delocalization for quasiperiodic operators with potentials of semi-bounded variation","authors":"Svetlana Jitomirskaya, Ilya Kachkovskiy","doi":"arxiv-2408.16935","DOIUrl":"https://doi.org/arxiv-2408.16935","url":null,"abstract":"We obtain the sharp arithmetic Gordon's theorem: that is, absence of\u0000eigenvalues on the set of energies with Lyapunov exponent bounded by the\u0000exponential rate of approximation of frequency by the rationals, for a large\u0000class of one-dimensional quasiperiodic Schr\"odinger operators, with no\u0000(modulus of) continuity required. The class includes all unbounded monotone\u0000potentials with finite Lyapunov exponents and all potentials of bounded\u0000variation. The main tool is a new uniform upper bound on iterates of cocycles\u0000of bounded variation.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179298","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 work is concerned with the spectral analysis of a piezoelectric energy harvesting model based on a coupled bending-torsion beam. After building the problem's operator setting and showing that the governing operator is nonselfadjoint with a purely discrete spectrum, we derive an asymptotic approximation of its spectrum. In doing so, we also prove that the addition of energy harvesting can be viewed as a weak perturbation of the underlying beam dynamics, in the sense that no piezoelectric parameters appear in the spectral approximation's first two orders of magnitude. We conclude by outlining future work based on numerical simulations.
{"title":"Spectral analysis of a coupled bending-torsion beam energy harvester: asymptotic results","authors":"Chris Vales","doi":"arxiv-2408.15635","DOIUrl":"https://doi.org/arxiv-2408.15635","url":null,"abstract":"This work is concerned with the spectral analysis of a piezoelectric energy\u0000harvesting model based on a coupled bending-torsion beam. After building the\u0000problem's operator setting and showing that the governing operator is\u0000nonselfadjoint with a purely discrete spectrum, we derive an asymptotic\u0000approximation of its spectrum. In doing so, we also prove that the addition of\u0000energy harvesting can be viewed as a weak perturbation of the underlying beam\u0000dynamics, in the sense that no piezoelectric parameters appear in the spectral\u0000approximation's first two orders of magnitude. We conclude by outlining future\u0000work based on numerical simulations.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179302","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 eigenvalue sums of Schr"odinger operators $-Delta+V$ on $L^2(R^d)$ with complex radial potentials $Vin L^q(R^d)$, $q
我们考虑了L^2(R^d)$上具有复径向势$Vin L^q(R^d)$, $q
{"title":"Bounds for Eigenvalue Sums of Schrödinger Operators with Complex Radial Potentials","authors":"Jean-Claude Cuenin, Solomon Keedle-Isack","doi":"arxiv-2408.15783","DOIUrl":"https://doi.org/arxiv-2408.15783","url":null,"abstract":"We consider eigenvalue sums of Schr\"odinger operators $-Delta+V$ on\u0000$L^2(R^d)$ with complex radial potentials $Vin L^q(R^d)$, $q<d$. We prove\u0000quantitative bounds on the distribution of the eigenvlaues in terms of the\u0000$L^q$ norm of $V$. A consequence of our bounds is that, if the eigenvlaues\u0000$(z_j)$ accumulates to a point in $(0,infty)$, then $(im z_j)$ is summable.\u0000The key technical tools are resolvent estimates in Schatten spaces. We show\u0000that these resolvent estimates follow from spectral measure estimates by an\u0000epsilon removal argument.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179300","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 H"older continuity of the Lyapunov exponent $L(omega,E)$ and the integrated density of states at energies that satisfy $L(omega,E)>4kappa(omega,E)cdot beta(omega)geq 0$ for general analytic potentials, with $kappa(omega,E)$ being Avila's acceleration.
{"title":"Hölder continuity of the integrated density of states for Liouville frequencies","authors":"Rui Han, Wilhelm Schlag","doi":"arxiv-2408.15962","DOIUrl":"https://doi.org/arxiv-2408.15962","url":null,"abstract":"We prove H\"older continuity of the Lyapunov exponent $L(omega,E)$ and the\u0000integrated density of states at energies that satisfy\u0000$L(omega,E)>4kappa(omega,E)cdot beta(omega)geq 0$ for general analytic\u0000potentials, with $kappa(omega,E)$ being Avila's acceleration.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179301","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}
Artur Bille, Victor Buchstaber, Pavel Ievlev, Svyatoslav Novikov, Evgeny Spodarev
The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in textit{graphene} and textit{carbon $(p,q)$--nanotubes}. The chemical and physical attributes of these two carbon allotropes are closely linked to the average number of closed paths of different lengths $kinmathbb{N}_0$ on their respective graph representations. Considering that a carbon $(p,q)$--nanotube can be thought of as a graphene sheet rolled up in a matter determined by the textit{chiral vector} $(p,q)$, our findings are based on the study of textit{random eigenvalues} of both the hexagonal and triangular lattices presented in cite{bille2023random}. This study reveals that for any given textit{chiral vector} $(p,q)$, the sequence of counts of closed paths forms a moment sequence derived from a functional of two independent uniform distributions. Explicit formulas for key characteristics of these distributions, including probability density function (PDF) and moment generating function (MGF), are presented for specific choices of the chiral vector. Moreover, we demonstrate that as the textit{circumference} of a $(p,q)$--nanotube approaches infinity, i.e., $p+qrightarrow infty$, the $(p,q)$--nanotube tends to converge to the hexagonal lattice with respect to the number of closed paths for any given length $k$, indicating weak convergence of the underlying distributions.
{"title":"Random eigenvalues of nanotubes","authors":"Artur Bille, Victor Buchstaber, Pavel Ievlev, Svyatoslav Novikov, Evgeny Spodarev","doi":"arxiv-2408.14313","DOIUrl":"https://doi.org/arxiv-2408.14313","url":null,"abstract":"The hexagonal lattice and its dual, the triangular lattice, serve as powerful\u0000models for comprehending the atomic and ring connectivity, respectively, in\u0000textit{graphene} and textit{carbon $(p,q)$--nanotubes}. The chemical and\u0000physical attributes of these two carbon allotropes are closely linked to the\u0000average number of closed paths of different lengths $kinmathbb{N}_0$ on their\u0000respective graph representations. Considering that a carbon $(p,q)$--nanotube\u0000can be thought of as a graphene sheet rolled up in a matter determined by the\u0000textit{chiral vector} $(p,q)$, our findings are based on the study of\u0000textit{random eigenvalues} of both the hexagonal and triangular lattices\u0000presented in cite{bille2023random}. This study reveals that for any given\u0000textit{chiral vector} $(p,q)$, the sequence of counts of closed paths forms a\u0000moment sequence derived from a functional of two independent uniform\u0000distributions. Explicit formulas for key characteristics of these\u0000distributions, including probability density function (PDF) and moment\u0000generating function (MGF), are presented for specific choices of the chiral\u0000vector. Moreover, we demonstrate that as the textit{circumference} of a\u0000$(p,q)$--nanotube approaches infinity, i.e., $p+qrightarrow infty$, the\u0000$(p,q)$--nanotube tends to converge to the hexagonal lattice with respect to\u0000the number of closed paths for any given length $k$, indicating weak\u0000convergence of the underlying distributions.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"153 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179303","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：