Spectrum asymptotics of a perturbed Laplace-Beltrami operator on a three-dimensional sphere
type: Item
instance of: scholarly article
title: Spectrum asymptotics of a perturbed Laplace-Beltrami operator on a three-dimensional sphere
zbMATH Open document ID: 1242.47037
DOI: 10.1134/S1064562411070258
published in: Doklady Mathematics
publication date: 2012-05-10T00:00:00Z
review text: This paper studies the asymptotics of the eigenvalues of the operator \(H=H_0 +V\) in \(L^2(S^3)\), where \(S^3\) is the unit three-dimensional sphere. Here, \(H_0\) is the Laplace-Beltrami operator and \(V\) is the operator of multiplication by a measurable real-valued function \(V(\omega)\in C(S^3)\). The spectrum of the operator \(H_0\) is discrete and consists of the eigenvalues \(\lambda_n = n(n + 2)\) of multiplicity \((n + 1)^2\). The main result of the paper is an asymptotic formula for \( \sum\limits_{k=1}^{(n+1)^2} \mu_n^{(k)}\), where \(\{ \mu_n^{(k)} \}_{k=1}^{(n+1)^2}\) is the set of eigenvalues of \(H\) contained in the interval \([\lambda_n - V_0, \lambda_n + V_0]\), where \(V_0=\max\limits_{\omega\in S^3}|V(\omega)|\). The proof is based on the investigation of the properties of the unperturbed resolvent \((H_0-z)^{-1}\).
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 6032766
zbMATH Keywords: spectrum asymptotics
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: three-dimensional sphere
zbMATH Keywords: perturbation
MaRDI profile type: Publication
cites work: Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere \(\mathbb{S}^2\)
author: Kh. Kh. Murtazin
author: A. I. Atnagulov
description: scientific article; zbMATH DE number 6032766