Capacity estimates for the scattering phase
type: Item
instance of: scholarly article
title: Capacity estimates for the scattering phase
zbMATH Open document ID: 0939.35131
author: Michael Demuth
author: I. McGillivray
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-12-05T00:00:00Z
review text: Let \(H\) be the selfadjoint nonnegative definite operator \(H:=-\frac 12\Delta\) acting in \(L^6({\mathbb R}^n)\) where \(n\geq 3\). The authors perturb \(H\) by imposing Dirichlet boundary conditions on a given compact subset \(K\) of \({\mathbb R}^N\). This gives a selfadjoint nonnegative definite operator \(H_Y\) acting in \(L^2(Y)\), where \(Y\) is the complement of \(K\). Let \(\theta_Y(\lambda)\) be the scattering phase for the scattering system \(\{H_Y, H;J\}\), where \(J\) denotes the restriction to \(Y\). The authors show that the variation in \(\theta_Y(\lambda)\) (modulo integer multiples of \(\pi /2\)) as \(K\) varies may be estimated in terms of the capacity of the symmetric difference of the obstacles. Furthermore, for the case where \(K\) has smooth boundary, \(H_{\beta}:=H+\beta 1_{k}\) (\(\beta >0\)) and \(\theta_{\beta}(\lambda)\) is the scattering phase for the scattering system \(\{ H_{\beta}, H\}\), the authors prove that \(\theta_{\beta}(\lambda)\) converges to \(\theta_{Y}(\lambda)\) (again modulo integer multiples of \(\pi /2\)) roughly at the rate \(\beta^{-1/2}\) as \(\beta \to \infty.\) A proof that \(\theta_{\beta}(\lambda)\) converges to \(\theta_{Y}(\lambda)\) as \(\beta\to\infty\) was sketched in [\textit{T. Kato}, Hadronic J. 1, 134-154 (1978; Zbl 0426.47004)].
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1348147
zbMATH Keywords: capacity estimates
zbMATH Keywords: scattering phase
zbMATH Keywords: scattering system
zbMATH Keywords: resolvent estimates
zbMATH Keywords: scattering matrix
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6478
OpenAlex ID: W2038215879
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DOI: 10.1006/JMAA.1999.6478
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description: scientific article; zbMATH DE number 1348147