Rank one perturbations in a Pontryagin space with one negative square
type: Item
instance of: scholarly article
title: Rank one perturbations in a Pontryagin space with one negative square
zbMATH Open document ID: 1028.47026
published in: Journal of Functional Analysis
publication date: 2002-07-04T00:00:00Z
review text: The interest in extending the study of rank one perturbations from Hilbert to Pontryagin spaces is justified by some recent results concerning the point interactions in quantum mechanical models. The framework of the present paper is a Pontryagin space \(({\mathfrak H},[.,.])\) of negative index one. The perturbations of a selfadjoint operator \(A\) on \({\mathfrak H}\) have the form \(A(t)=A+ t[.,\omega]\omega\), where \(w\in{\mathcal H}\) and \(t\in \mathbb{R}\). The spectral analysis of these perturbations is based on a subclass \(N_{0,1}\) of the generalized Nevanlinna functions, for which \(\lim_{y\to \infty} Q(iy)/y=0\) and \(\limsup_{y\to\infty} y|\text{Im} Q(iy)|<\infty\). Most results refer to the explicit characterization of the critical eigenvalues of \(A(t)\) and to the root subspaces of the Friedrichs extension \(S_F=S_+(\{0\} \oplus\text{span}\{\omega\})\), where \(S\) is the restriction of \(A\) to \(\text{dom} S =\{f\in\text{dom} A:[f,\omega]= 0\}\). The last section of the paper contains examples encountered in mathematical physics.
reviewed by: Trandafir Balan
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 46N50
zbMATH DE Number: 1763776
zbMATH Keywords: Nevanlinna functions
zbMATH Keywords: rank one perturbations
zbMATH Keywords: Pontryagin space
zbMATH Keywords: critical eigenvalues
zbMATH Keywords: root subspaces
zbMATH Keywords: Friedrichs extension
author: V. A. Derkach
author: S. Hassi
author: H. S. V. de Snoo
MaRDI profile type: Publication
full work available at URL: jfan.2001.3837
OpenAlex ID: W2103055410
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DOI: 10.1006/JFAN.2001.3837
description: scientific article; zbMATH DE number 1763776