Complex transmission eigenvalues for spherically stratified media
type: Item
instance of: scholarly article
title: Complex transmission eigenvalues for spherically stratified media
zbMATH Open document ID: 1260.47012
DOI: 10.1088/0266-5611/28/7/075005
author: Yuk-J Leung
author: David Colton
published in: Inverse Problems
publication date: 2012-08-24T00:00:00Z
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 78A46
zbMATH DE Number: 6071992
zbMATH Keywords: transmission eigenvalues
zbMATH Keywords: inverse scattering problem
zbMATH Keywords: spherically stratified index of refraction
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1088/0266-5611/28/7/075005
OpenAlex ID: W1991943387
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review text: The transmission eigenvalue problem for a spherically stratified medium in \(\mathbb{R}^d\), \(d=2,3\), is to find a nontrivial solution \(v,w\) of NEWLINE\[NEWLINE\Delta w+k^2 \eta w =0,\;\Delta v+k^2v = 0 \text{ in } B; \;w=v, \partial_\nu w = \partial_\nu v \text{ on } \partial B , \tag{1}NEWLINE\]NEWLINE where \(B = \{| x| < 1\}\) and the index of refraction \(\eta\) only depends on \(| x|\). When \(v\) and \(w\) only depend on \(| x|\), (1) reduces to a coupled set of ordinary differential equations. In this case, the authors prove the following results on the existence of complex transmission eigenvalues. For \(d=2\) and a constant index of refraction, there is an infinite number of complex eigenvalues. For \(d=3\) and constant \(\eta \in \mathbb{N}\), there are no complex eigenvalues, whereas such eigenvalues can exist if \(\eta\) is a rational number. Moreover, complex eigenvalues may exist for a spherically stratified variable index of refraction.
reviewed by: Johannes Elschner
description: scientific article; zbMATH DE number 6071992