Scattering problem in the theory of singular perturbations of self-adjoint operators
type: Item
instance of: scholarly article
title: Scattering problem in the theory of singular perturbations of self-adjoint operators
zbMATH Open document ID: 0555.47009
DOI: 10.1007/BF01057464
author: Hagen Neidhardt
author: Volodymyr Koshmanenko
author: Manfred Wollenberg
published in: Ukrainian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let \(H_ 0>0\) be a selfadjoint operator and let H be a perturbed operator of \(H_ 0\) with singular perturbation. Suppose that \(H_ 0\) and H act in different Hilbert spaces \({\mathfrak H}\) and \({\mathfrak H}'\), where the identification operator J:\({\mathfrak H}\to {\mathfrak H}'\), defined on a dense set \(\Phi\), is not necessarily bounded. This paper is devoted to the study of the corresponding wave operator \(W_{\pm}(H_ 0,H,J,D)=W_{\pm},\) where D is a fixed subset of \(\Phi\). It contains many results without proofs about the existence and closedness of \(W_{\pm}\), the existence of the wave operator \(W_{\pm}(\phi (H_ 0),\phi (H),J,D)=W_{\pm \phi}\) for an admissible function \(\phi\), and about the question when \(W_{\pm \phi}H_{\pm}\) holds. Finally, the applicability of the nuclear methods and the method of orthogonal extension are considered.
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 3884802
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: wave operator
zbMATH Keywords: nuclear methods
zbMATH Keywords: method of orthogonal extension
reviewed by: Ivan Vidav
MaRDI profile type: Publication
cites work: Q3938067
cites work: Scattering theory with two Hilbert spaces
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description: scientific article; zbMATH DE number 3884802