Eigenfrequencies and eigenfunctions of the Laplacian for Neumann boundary conditions in a system of two coupled cavities
type: Item
instance of: scholarly article
title: Eigenfrequencies and eigenfunctions of the Laplacian for Neumann boundary conditions in a system of two coupled cavities
zbMATH Open document ID: 0853.35084
DOI: 10.1007/BF01018571
published in: Theoretical and Mathematical Physics
publication date: 1996-08-12T00:00:00Z
review text: We consider a region \(\Omega\) in \(\mathbb{R}^2\) that consists of two regions \(\Omega_1\) and \(\Omega_2\) with boundaries belonging to \(C^2\) and connected by a thin rectangular channel \(P\) of width \(\Omega\). Our task is to investigate the eigenvalues and eigenfunctions of the Laplacian for Neumann boundary conditions in the region \(\Omega\). We analyze a model problem in which the role of the channel is played by a line segment, and the coupling between the regions is specified by means of the technique of the theory of extensions of symmetric operators. We find the first terms in the asymptotic expansions of the eigenvalues and eigenfunctions of a model operator in the region \(\Omega\) with respect to the parameter \(\alpha\), which specifies the strength of the coupling between the regions, and we obtain an expression for the resolvent in the form of a series in powers of \(\alpha\).
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 916202
zbMATH Keywords: asymptotic expansions of the eigenvalues
author: Alexander Kiselev
author: Boris S. Pavlov
MaRDI profile type: Publication
cites work: The essential spectrum of Neumann Laplacians on some bounded singular domains
cites work: Eigenvalues and resonances for domains with tubes: Neumann boundary conditions
description: scientific article; zbMATH DE number 916202