Lower bounds for resonance widths in potential and obstacle scattering
type: Item
instance of: scholarly article
title: Lower bounds for resonance widths in potential and obstacle scattering
zbMATH Open document ID: 0712.35072
DOI: 10.1007/BF02108782
author: Claudio Fernández
author: Richard B. Lavine
published in: Communications in Mathematical Physics
publication date: 1990-01-01T00:00:00Z
review text: Consider the Schrödinger equation \((i\partial_ t+\Delta -V(x))\psi =0\), \(x\in {\mathbb{R}}^ 3\) and the wave equation \((\partial_ t^ 2- \Delta)u=0\) outside an obstacle \(\Omega\) in \({\mathbb{R}}^ 3\). It is assumed in both cases that the perturbation is supported in a ball \(B_{\rho}.\) The authors obtain an explicit lower bound for the size of the imaginary parts of the resonances \(k=\kappa -i\eta\) for this equation. The first main result deals with the Schrödinger operator at energies where classical trapping does not happen. It is shown that potentials supported in a ball \(B_{\rho}\), satisfying \(\kappa^ 2-V-r\partial V/\partial r>0\) do not produce resonances with \(\eta\) R very small and \(\kappa\) R not too small. For \(\kappa\) large condition becomes approximately \(4.72\eta R<1.\) The second result deals with potentials for which the non-trapping condition fails. Then resonances with \(\eta R\ll 1\) are expected. The authors give lower bounds which are exponentially small in a certain quantity which roughly measures the size of the trapping barrier. Finally the authors give a lower bound of \(\eta\) for the Laplacian in the exterior of a star-shaped obstacle. This lower bound approaches \(R^{- 1}(2+e)^{-1}\) as \(\kappa\to \infty\).
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 81U20
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 4171520
zbMATH Keywords: resonance widths
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: resonances
zbMATH Keywords: trapping
zbMATH Keywords: potentials
zbMATH Keywords: star-shaped obstacle
reviewed by: Plamen Stefanov
MaRDI profile type: Publication
cites work: Scattering frequencies and Gevrey 3 singularities
cites work: On the location of resonances for Schrödinger operators in the semiclassical limit. I: Resonances free domains
cites work: Q3788791
cites work: General lower bounds for resonances in one dimension
cites work: Q5620043
cites work: Q5651106
cites work: Decay of solutions of the wave equation outside nontrapping obstacles
cites work: The first variation of the scattering matrix: An addendum
cites work: Q4185363
cites work: Sharp exponential bounds on resonances states and width of resonances
full work available at URL: https://doi.org/10.1007/bf02108782
OpenAlex ID: W2088177701
description: scientific article; zbMATH DE number 4171520