Perturbation of an eigenvalue in the continuous spectrum of a waveguide with an asymmetric obstacle
type: Item
instance of: scholarly article
title: Perturbation of an eigenvalue in the continuous spectrum of a waveguide with an asymmetric obstacle
zbMATH Open document ID: 1234.35165
DOI: 10.1134/S1064562411060202
published in: Doklady Mathematics
publication date: 2012-01-06T00:00:00Z
review text: The author considers a two-dimensional infinitely long waveguide formed by two parallel lines with an asymmetric obstacle. The problem is mathematically related to an eigenvalue problem of the Laplace operator in the corresponding domain with Neumann boundary conditions. The spectrum of the eigenvalue problem is continuous. The eigenvalues fill the closed real half line \([0, \infty)\). Additionally, a point spectrum may exist which is associated with trapped modes. \textit{D. V. Evans, M. Levitin}, and \textit{D. Vassiliev} proved in [J. Fluid Mech. 261, 21--31 (1994; Zbl 0804.76075)] the existence of at least one trapped mode, which provokes resonance phenomena, for the case of obstacles which are symmetric about the centreline of the waveguide. Based on the method used in this article and imposing artificial Dirichlet boundary conditions on the midline of the waveguide with the symmetric obstacle, the author showed in [Sib. Math. J. 51, No. 5, 866--878 (2010); translation from Sib. Mat. Zh. 51, 1086--1101 (2010; Zbl 1205.35178)] that the problem, under certain inequality condition, has an eigenvalue in the interval \((0, {\pi}^2)\) of the continuous spectrum. These results and the finding in [\textit{A. Aslanyan, L. Parnovski} and \textit{D. Vassiliev}, Q. J. Mech. Appl. Math. 53, No. 3, 429--447 (2000; Zbl 0972.76091)] that the shift of the symmetric obstacle by a small distance along the vertical axis (that is, changeover to an asymmetric obstacle) gives a complex eigenvalue, have led to the belief that the formation of the eigenvalue in the continuous spectrum requires certain symmetry. In the present article it is shown, using an asymptotic representation of eigenvalues inside a continuous spectrum, that the eigenvalue persists in the continuous spectrum if the perturbed boundary of the obstacle satisfies certain integral conditions. This incomplete stability of the point spectrum is called forced stability.
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 78A50
zbMATH DE Number: 5995578
zbMATH Keywords: waveguide
zbMATH Keywords: asymmetric obstacle
zbMATH Keywords: Laplace operator
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: eigenvalue problem, eigenvalue perturbation
zbMATH Keywords: perturbed obstacle boundary
zbMATH Keywords: continuous spectrum
zbMATH Keywords: point spectrum
zbMATH Keywords: forced stability
author: Sergei A. Nazarov
MaRDI profile type: Publication
cites work: Existence theorems for trapped modes
cites work: Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold
cites work: Complex resonances in acoustic waveguides
cites work: Embedded trapped modes in water waves and acoustics
cites work: Elliptic problems in domains with piecewise smooth boundaries
cites work: [Russian Text Ignored.]
full work available at URL: s1064562411060202
OpenAlex ID: W2004342788
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description: scientific article; zbMATH DE number 5995578