Asymptotics of the distribution of eigenvalues of a selfadjoint second order hyperbolic differential operator on the two-dimensional torus
type: Item
instance of: scholarly article
title: Asymptotics of the distribution of eigenvalues of a selfadjoint second order hyperbolic differential operator on the two-dimensional torus
zbMATH Open document ID: 1204.47061
author: V. M. Kaplitskiĭ
published in: Siberian Mathematical Journal
publication date: 2010-12-08T00:00:00Z
review text: The author studies a class of general selfadjoint second order hyperbolic operators \(A\) on the two-dimensional torus \(T^2\) with sufficiently smooth coefficients under some conditions that guarantee that the spectrum is discrete. The method consists in constructing a regularizer for the hyperbolic operator and reducing the problem to studying the spectrum of a linear operator pencil, for which the spectrum of the principal unperturbed part is computed explicitly, and the perturbation does not affect the principal term of the asymptotics of \(N(\lambda ;A):=\)card\(\{j: |\lambda_j(A)|\leq\lambda\}\) as \(\lambda\rightarrow +\infty\). Explicit calculation of the asymptotics for the unperturbed pencil reduces to the classical problem of finding the asymptotics for the number of integer points in the domain bounded by a hyperbola as the domain extends indefinitely.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5824086
zbMATH Keywords: hyperbolic operator
zbMATH Keywords: distribution of eigenvalues
zbMATH Keywords: spectrum
MaRDI profile type: Publication
full work available at URL: s11202-010-0084-6
OpenAlex ID: W2154641220
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DOI: 10.1007/S11202-010-0084-6
description: scientific article; zbMATH DE number 5824086