Scattering theory in a pair of spaces
type: Item
instance of: scholarly article
title: Scattering theory in a pair of spaces
zbMATH Open document ID: 0616.47009
DOI: 10.1007/BF01077288
author: Dmitri Yafaev
publication date: 1986-01-01T00:00:00Z
review text: Some elementary notions of two-space scattering theory are discussed. It is shown that for the pair of self-adjoint operators \(H_ 0\), H and identification J the existence of the (strong) wave operator \(W_{\pm}(H,H_ 0;J)\) is equivalent to the existence of weak wave operators \(\tilde W_{\pm}(H,H_ 0;J)\), \(\tilde W_{\pm}(H_ 0,H_ 0;J^*J)\) and to the equality \[ \tilde W_{\pm}(H,H_ 0;J)^*\tilde W_{\pm}(H,H_ 0;J)=\tilde W_{\pm}(H_ 0,H_ 0;J^*J). \] New definition of the completeness of \(W_{\pm}(H,H_ 0;J)\) is proposed. It is related to the asymptotics of exp(iHt) as \(t\to \pm \infty\) and depends on J. Let \(W_{\pm}(H,H_ 0;J)\) exist. Then \(W_{\pm}(H_ 0,H;J^*)\) is shown to exist iff \(W_{\pm}(H,H_ 0;J)\) is complete and \(W_{\pm}(H_ 0,H_ 0;J^*J)\) also exists.
Mathematics Subject Classification ID: 47A40
zbMATH DE Number: 3998563
zbMATH Keywords: two-space scattering theory
zbMATH Keywords: wave operator
zbMATH Keywords: completeness
MaRDI profile type: Publication
cites work: Scattering theory with two Hilbert spaces
cites work: Q4191141
cites work: Q3320954
full work available at URL: https://doi.org/10.1007/bf01077288
OpenAlex ID: W2060107176
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3998563