Weyl's law for Neumann Schrödinger operators on Hölder domains
type: Item
instance of: scholarly article
title: Weyl's law for Neumann Schrödinger operators on Hölder domains
zbMATH Open document ID: 1546.35129
author: Charlotte Dietze
published in: Séminaire Laurent Schwartz. EDP et Applications
publication date: 2024-07-03T00:00:00Z
review text: In this survey paper, based on the author's paper [``Semiclassical estimates for Schrödinger operators with Neumann boundary conditions on Hölder domains'', Preprint, \url{arXiv:2304.01587}], the main points of her own results on the Weyl asymptotic expansion formula for the number of eigenvalues of a Schrödinger operator defined on a bounded \(\gamma\)-Hölder domain under Neumann boundary conditions are given. As a byproduct, it also explains that another proof of the Weyl asymptotic expansion formula for the Neumann Laplacian on a bounded \(\gamma\)-Hölder domain obtained by \textit{Y. Netrusov} and \textit{Y. Safarov} [Commun. Math. Phys. 253, No. 2, 481--509 (2005; Zbl 1076.35085)] can be obtained.\N\NFirst, the results of Rayleigh, Sommerfeld, Lorentz, Weyl, Rozenblum, Kac and Ivrii, etc. about the famous Weyl asymptotic expansion formula for the Dirichlet Laplacian are mentioned. Next, it is explained that the situation for the Neumann Laplacian is quite different from that for the Dirichlet Laplacian, citing the Hempel-Seco-Simon result [\textit{R. Hempel} er al., J. Funct. Anal. 102, No. 2, 448--483 (1991; Zbl 0741.35043)] that zero belongs to the essential spectrum in case of the Neumann Laplacian.\N\NThen, Netrusov-Safarov's Neumann-Laplacian results are introduced. That is, under the assumption that a bounded open set \(\Omega\subset\mathbb{R}^d\) (\(d\geqq2\)) is a \(\gamma\)-Hölder domain, two results are shown: \((i)\) the Weyl asymptotic expansion formula holds for all \(\gamma\) satisfying \(\frac{d-1}{d}<\gamma<1\), and \((ii)\) if \(\gamma\) satisfies \(0<\gamma\leqq \frac{d-1}{d}\), then there exists a \(\gamma\)-Hölder domain \(\Omega\) that does not satisfy the Weyl asymptotic expansion formula. In this theorem, \(\Omega\) is called a \(\gamma\)-Hölder domain if it is a domain and \(\partial\Omega\), the boundary of \(\Omega\), is locally the graph of a \(\gamma\)-Hölder continuous function \(f\) satisfying \(\left|f(x)-f(y)\right|\leqq c\left|x…
reviewed by: Hideo Nakazawa
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 7875005
zbMATH Keywords: Neumann Laplacian
zbMATH Keywords: Holder domains
zbMATH Keywords: Cwikel-Lieb-Rozenblum inequality
zbMATH Keywords: semiclassical asymptotics
zbMATH Keywords: Weyl's law
MaRDI profile type: Publication
cites work: Q3633715
cites work: Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik.
cites work: Weak type estimates for singular values and the number of bound states of Schrödinger operators
cites work: Über quadratische Formen mit reellen Koeffizienten.
cites work: Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain
cites work: Equivalence of Sobolev inequalities and Lieb-Thirring inequalities
cites work: Isospectral plane domains and surfaces via Riemannian orbifolds
cites work: The essential spectrum of Neumann Laplacians on some bounded singular domains
cites work: Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary
cites work: Can One Hear the Shape of a Drum?
cites work: Embedding of Sobolev spaces on Hölder domains
cites work: Bounds on the eigenvalues of the Laplace and Schroedinger operators
cites work: Alte und neue Fragen der Physik.
cites work: EIGENVALUES OF THE LAPLACE OPERATOR ON CERTAIN MANIFOLDS
cites work: Weyl asymptotic formula for the Laplacian on domains with rough boundaries
cites work: Q5838364
cites work: The dynamical theory of gases and of radiation.
cites work: ON THE EIGENVALUES OF THE FIRST BOUNDARY VALUE PROBLEM IN UNBOUNDED DOMAINS
cites work: Die \textit{Green}sche Funktion der Schwingungsgleichung für ein beliebiges Gebiet.
cites work: Über die asymptotische Verteilung der Eigenwerte.
cites work: Inverse spectral problem for analytic domains. II: \(\mathbb Z_2\)-symmetric domains
DOI: 10.5802/SLSEDP.158
description: scientific article; zbMATH DE number 7875005