On a framework of scattering for dissipative systems
type: Item
instance of: scholarly article
title: On a framework of scattering for dissipative systems
zbMATH Open document ID: 1037.35052
author: Mitsuteru Kadowaki
published in: Osaka Journal of Mathematics
publication date: 2003-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1153493046
review text: Let \({\mathcal H}\) be a separable Hilbert space. Let \(A\) be the generator of a contraction semigroup \(\{V(t)\}_{t\geq 0}\) in \({\mathcal H}\) and \(\{U_0(t)\}_{t\in\mathbb{R}}\) be an unitary group in \({\mathcal H}\). Let \({\mathcal H}_b\) be the space generated by the eigenvectors of \(A\) with real eigenvalues. Under some assumptions, the author proves the existence of the wave operator \[ wf= \lim_{t\to\infty} U_0(t) V(t) f,\quad f\in{\mathcal H}^1_b. \] As a corollary, he finds scattering solutions of \(d(V(t)f)/dt= -iAV(t)f\), \(f\in D(A)\). He also obtains standard results concerning real eigenvalues of \(A\): Any non-zero eigenvalue has finite multiplicity and the possible finite accumulation point of the real eigenvalues of \(A\) is zero. As applications, one considers elastic wave equation with dissipative boundary condition in a half space of \(\mathbb{R}^3\), the Schrödinger equation with complex valued potential and acoustic wave equation with dissipative term in inhomogeneous media.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 74J20
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 1894636
zbMATH Keywords: contraction semigroup
zbMATH Keywords: real eigenvalues
zbMATH Keywords: wave operator
zbMATH Keywords: scattering solutions
zbMATH Keywords: elastic wave equation
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: acoustic wave equation
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
Recommended article: Q4010062
Recommended article: Scattering Problems for Symmetric Systems with Dissipative Boundary Conditions
Recommended article: Scattering for solutions of a dissipative nonlinear Schrödinger equation
Recommended article: Q3973537
Recommended article: Scattering matrices for dissipative quantum systems
Recommended article: Trace Formulae for Dissipative and Coupled Scattering Systems
Recommended article: On the dissipative Lax-Phillips scattering theory
Recommended article: Scattering of traveling spots in dissipative systems
Recommended article: Stable Dissipative Linear Stationary Dynamical Scattering Systems
Recommended article: Resolvent estimates and scattering states for dissipative systems.
description: scientific article; zbMATH DE number 1894636