Trapping obstacles with a sequence of poles on the scattering matrix converging to the real axis
type: Item
instance of: scholarly article
title: Trapping obstacles with a sequence of poles on the scattering matrix converging to the real axis
zbMATH Open document ID: 0617.35102
author: Mitsuru Ikawa
published in: Osaka Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The scattering of the acoustic equation by bounded obstacles is considered. In the book of \textit{P. D. Lax} and \textit{R. S. Phillips} [''Scattering theory'' (1967; Zbl 0186.163)], they conjectured that, when the obstacle is trapping, the scattering matrix \({\mathcal S}(z)\) has a sequence of poles converging to the real axis. Although it is believed since a long time that this conjecture is correct, we didn't find even one of such examples. The purpose of this paper is to give such an example of obstacle. Suppose that \({\mathcal O}\) satisfies: (i) \({\mathcal O}={\mathcal O}_ 1\cup {\mathcal O}_ 2\), where \({\mathcal O}_ 1\) and \({\mathcal O}_ 2\) are convex and bounded in \({\mathbb{R}}^ 3\) such that \(\bar {\mathcal O}_ 1\cap \bar {\mathcal O}_ 2=\phi.\) (ii) Let \(a_ j\in \Gamma_ j=\partial {\mathcal O}_ j\), \(j=1,2\), be the points such that \(| a_ 1-a_ 2| =dis({\mathcal O}_ 1,{\mathcal O}_ 2)\). Then it holds that \[ C| x-a_ j|^ e \geq \kappa_{j\ell}(x)\geq C^{-1}| x-a_ j|^ e\text{ for all } x\in \Gamma_ j \] for some \(2\leq e<\infty\), where \(\kappa_{j\ell}\), \(\ell =1,2\) are the principal curvatures of \(\Gamma_ j\) at \(\kappa\). Then the scattering matrix \({\mathcal S}(z)\) has a sequence of the poles \(\{z_ j\}^{\infty}_{j=1}\) such that Im \(z_ j\to 0\), \(j\to \infty\).
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 4000570
zbMATH Keywords: trapping obstacle
zbMATH Keywords: scattering
zbMATH Keywords: acoustic equation
zbMATH Keywords: bounded obstacles
zbMATH Keywords: poles
zbMATH Keywords: scattering matrix
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4000570