On the perturbation theory of self-adjoint operators
type: Item
instance of: scholarly article
title: On the perturbation theory of self-adjoint operators
zbMATH Open document ID: 1182.47013
author: L. P. Nizhnik
author: S. Albeverio
author: Sergii Kuzhel
published in: Tokyo Journal of Mathematics
publication date: 2009-04-21T00:00:00Z
review text: Let \(A\) be a semi-bounded selfadjoint operator on a separable Hilbert space \(\mathcal{H}\). A selfadjoint operator \(\tilde{A} \neq A\) on \(\mathcal{H}\) is called regular perturbation of \(A\) if \(D(A) \cap D(\tilde{A}) =D(A)\) and singular perturbation otherwise, where \(D(A)\) denotes the domain of the operator. Furthermore, a singular perturbation \(\tilde{A}\) is said to be purely singular if the maximal common part of \(A\) and \(\tilde{A}\) is densely defined, and mixed singular otherwise. Compared to mixed singular perturbations, regular and purely singular perturbations are well-established by the additive perturbation theory and the extension theory, respectively. In this paper, the authors show that all types of perturbations (regular, purely singular, and mixed singular) can be described and studied in both the frameworks of extension theory and the additive perturbation theory. Their method is to develop a bijection between selfadjoint operators \(\tilde{A}\) and the set of pairs \(\{N, T\}\), where \(N\) is a subspace of \(\mathcal{H}\) and \(T\) is an admissible selfadjoint operator in \(N\), and then to establish connection between \(\{N, T\}\) and the parameters in the frameworks of the extension theory and the additive perturbation theory. They also use the correspondence \(\tilde{A} \leftrightarrow \{N, T\}\) to show that any singular finite rank perturbation can be approximated by regular finite rank perturbations in the norm resolvent sense. Lastly, an application is given to the study of Schrödinger operators with point interactions.
reviewed by: Miyeon Kwon
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 5545412
zbMATH Keywords: selfadjoint perturbations
zbMATH Keywords: boundary value spaces
zbMATH Keywords: Krein's resolvent formula
zbMATH Keywords: additive perturbations
zbMATH Keywords: finite rank perturbations
zbMATH Keywords: resolvent approximation
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: point interactions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1233844052
OpenAlex ID: W1968944229
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DOI: 10.3836/TJM/1233844052
description: scientific article; zbMATH DE number 5545412