Spectral and scattering theory in perturbed stratified fluids
type: Item
instance of: scholarly article
title: Spectral and scattering theory in perturbed stratified fluids
zbMATH Open document ID: 0597.76074
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1985-01-01T00:00:00Z
review text: The author considers the spectral and scattering theory of the acoustic propagator of a perturbed stratified fluid. This involves the study of the operator \(A=-c^ 2(x,y)\Delta\); under suitable assumptions on the measurable function c, he proves that the spectrum of A coincides with the essential spectrum, and consists of [0,\(\infty)\). There is no singular continuous spectrum, the eigenvalues of A on \({\mathbb{R}}^+\setminus \{\cup_{k=1,\infty}\mu_ k\}\) have finite multiplicity and can only accumulate at zero and \(\{\) \(\cup_{k=1,\infty}\mu_ k\}\). The numbers \(\mu_ k\) are the thresholds at which a new guided wave of the unperturbed operator appears. Although this result is not completely new, the proof given by the author in the present paper is really simpler and definitely shorter than the known one. The generalized Fourier maps \(F_ A^{\pm}\) and the scattering operator S are then investigated. In particular it is proved that S is unitary and that the wave operators can be obtained as suitable strong limits of a function of both the perturbed and unperturbed acoustic operator.
Mathematics Subject Classification ID: 76Q05
Mathematics Subject Classification ID: 76V05
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 3963450
zbMATH Keywords: acoustic propagator
zbMATH Keywords: perturbed stratified fluid
zbMATH Keywords: essential spectrum
zbMATH Keywords: singular continuous spectrum
zbMATH Keywords: eigenvalues
zbMATH Keywords: generalized Fourier maps
zbMATH Keywords: scattering operator
author: Ricardo Weder
reviewed by: Fabio Rosso
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3963450