Scattering theory for perturbed stratified media
type: Item
instance of: scholarly article
title: Scattering theory for perturbed stratified media
zbMATH Open document ID: 0926.35106
DOI: 10.1007/BF02786928
author: K. Appert
published in: Journal d'Analyse Mathématique
publication date: 1999-11-29T00:00:00Z
review text: A trace formula for the operator of acoustic propagation in a perturbed stratified medium is proved. The operator of acoustic propagation in the stratified medium is of the form \(H_0= -c^2_0(y)/\mu_0(y) \nabla_z\mu^{-1}_0(y) \nabla_z\), \(z= (x,y)\in \mathbb{R}^n\times \mathbb{R}\), where the speed of propagation \(c_0\) and the density of the medium \(\mu_0\) are bounded positive piecewise smooth functions with a finite number of singularities and are rapidly convergent to some constants \(c_\pm\), \(\mu_\pm\), \(0<c_+\leq c_-\), when \(y\to\pm\infty\) in the sense that \(c_0(y)- c_\pm\), \(\partial^j_y(\mu_0(y)- \mu_\pm)\), \(j= 0,1\) converge to zero, when \(y\to\pm\infty\) faster than any negative power of \(| y|\). The operator of acoustic propagation in the perturbed stratified medium is \(H_0= -c^2(z)\mu(z)\nabla_z \mu^{-1}(z) \nabla_z\), where \(c- c_0\), \(\mu- \mu_0\) and \(|\nabla_z(\mu- \mu_0)|\) converge to zero when \(| z|\to\pm\infty\) faster than any negative power of \(| z|\). Other additional assumptions on \(c_0\), \(\mu_0\), \(c\) and \(\mu\), which we shall not mention here, are made. The proof of the trace formula is based on the knowledge of the asymptotic expansions of the generalized eigenfunctions of \(H\). A blow-up space is introduced in order to better describe this asymptotic expansions. The main section of the paper is devoted to the proof of asymptotic expansion formulas for the pullbacks to the blow-up space of \((H-(\lambda+ i0)^2)^{-1}f\), for \(f\) in \(L^2_{\text{comp}}(\mathbb{R}^{n+1})+ {\mathcal S}\left(\mathbb{R}^n; \bigcap_{k\in\mathbb{N}}(1+| y|^2)^{-k/2} L^2(\mathbb{R})\right)\). The asymptotic expansion formulas for the generalized eigenfunctions are obtained from this more general result. Then a relative scattering matrix \(S^r(\lambda)\) is defined and is related to the asymptotics of the generalized eigenfunctions. It is proved that the scattering matrix is unitary and of the form \(S^r(\lambda)= I+ A(\lambda)\) with …
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 1274822
zbMATH Keywords: asymptotic of generalized eigenfunctions
zbMATH Keywords: trace formula
zbMATH Keywords: scattering matrix
MaRDI profile type: Publication
cites work: Spectral theory of schrödinger operators on euclidean and on non-euclidean spaces
cites work: La relation de possion pour l'equation des ondes dans un ouvert non borne application a la theorie de la diffusion
cites work: Q3829912
cites work: Q5593418
cites work: Q4092387
cites work: Spectral and scattering theory for wave propagation in perturbed stratified media
cites work: Q5728680
cites work: Scattering theory for manifolds with asymptotically cylindrical ends
cites work: Spectral analysis for optical fibres and stratified fluids. I: The limiting absorption principle
cites work: Spectral analysis for optical fibres and stratifled fluids II: Absence of Eigenvalues
cites work: Inverse scattering on the line
cites work: Théorie spectrale de la progagation des ondes acoustiques dans un milieu stratifié perturbé. (Spectral analysis of the propagation of acoustic waves in a perturbed stratified medium)
cites work: The spectrum of positive elliptic operators and periodic bicharacteristics
cites work: The limiting absorption and amplitude principles for the diffraction problem with two unbounded media
cites work: Q4275389
cites work: Scattering asymptotics for Riemann surfaces
cites work: The spectral function of an elliptic operator
cites work: Q3996146
cites work: Asymptotic properties of generalized eigenfunctions for three body Schrödinger operators
cites work: Inverse scattering theory for wave equations in stratified media
cites work: Q5732462
cites work: Q5579836
cites work: Scattering Theory for Automorphic Functions. (AM-87)
cites work: Q4849528
cites work: Q4278406
cites work: Weyl asymptotics for the phase in obstacle scattering
cites work: Q4310183
cites work: Q4008616
cites work: Spectral geometry and scattering theory for certain complete surfaces of finite volume
cites work: Q5730343
cites work: Q4763639
cites work: Q3924605
cites work: Q4191141
cites work: On Sommerfeld radiation conditions for the diffraction problem with two unbounded media
cites work: Q3971025
cites work: Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du laplacien
cites work: Q4325902
cites work: Partial differential equations. 2: Qualitative studies of linear equations
cites work: Structure of the resolvent for three-body potentials
cites work: Asymptotic behavior of generalized eigenfunctions in \(N\)-body scattering
cites work: Q4941765
cites work: Sound propagation in stratified fluids
cites work: Q3134926
cites work: Reciprocity relations and completeness of far‐field pattern vectors for obstacle scattering of acoustic wave in a stratified medium
cites work: Radiation condition and scattering problem for time-harmonic acoustic waves in a stratified medium with a nonstratified inhomogeneity
cites work: Q3996935
full work available at URL: https://doi.org/10.1007/bf02786928
OpenAlex ID: W2032043817
description: scientific article; zbMATH DE number 1274822