A result on the scattering theory for first order systems with long-range perturbations
type: Item
instance of: scholarly article
title: A result on the scattering theory for first order systems with long-range perturbations
zbMATH Open document ID: 0661.35068
author: Koji Kikuchi
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: The author studies the differential equation \(D_ tu=\Lambda u\), where u is a \(C^ m\)-valued function of \(x\in R^ n\), \(D_ t=(1/i)(\partial /\partial t),\) \(\Lambda =E(x)^{-1/2}\sum^{n}_{j=1}A_ jD_ jE(x)^{-1/2},\) each \(A_ j\) is an m by m constant hermitian matrix, and E(x) is a continuous m by m hermitian matrix valued function with \(0<c_ 1I\leq E(x)\leq c_ 2I\). \(\Lambda\) can be extended to a selfadjoint operator on \({\mathcal H}=L^ 2(R^ n)\). \(\Lambda^ 0=\sum^{n}_{j=1}A_ jD_ j\) can also be extended to a selfadjoint operator on \({\mathcal H}\). In the present paper \(\Lambda\) is regarded as a perturbation of \(\Lambda^ 0\). The main result is the existence of a modified wave operator for \(\Lambda^ 0\) and \(\Lambda\) under the assumption that the perturbation is long-range, i.e., (1) \(E(x)\in C^{\infty}(R^ n)\) and (2) \(| \partial_ x^{\alpha}(E(x)- I)| \leq (1+| x|)^{-\delta -| \alpha |}\) for \(0<\delta \leq 1\) and \(| \alpha | \geq 0\). (In the case of a short- range perturbation, i.e., \(\delta >1\), it is already known that, for a wide class of \(\Lambda^ 0\), the wave operators exist and are complete.) The theory of the paper is applied to the Maxwell equation in crystals.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 47L90
zbMATH DE Number: 4080046
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: perturbation
zbMATH Keywords: existence
zbMATH Keywords: modified wave operator
zbMATH Keywords: long-range
zbMATH Keywords: short-range
zbMATH Keywords: Maxwell equation
zbMATH Keywords: crystals
reviewed by: Richard C. Gilbert
MaRDI profile type: Publication
full work available at URL: pjaa.63.179
OpenAlex ID: W1966227724
cites work: The existence of wave operators in scattering theory
cites work: Scattering theory for first order systems with long-range perturbations: The existence of the modified wave operators
cites work: Spectral and scattering theory for symmetric systems of nonconstant deficit
cites work: Spectral analysis of strongly propagative systems.
cites work: Electromagnetic signal propagation in crystals
DOI: 10.3792/PJAA.63.179
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description: scientific article; zbMATH DE number 4080046