Removing cut-offs from singular perturbations: An abstract result
type: Item
instance of: scholarly article
title: Removing cut-offs from singular perturbations: An abstract result
zbMATH Open document ID: 0563.47009
DOI: 10.1007/BF00420179
published in: Letters in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: Let \(H_ 0\geq 0\) and V be selfadjoint operators in a Hilbert space such that for some dense subspace D with \(D\subset D(H_ 0)\) and for \(0<a<1\) and \(b>0\), \[ \| Vf\| \leq a\| H_ 0f\| +b\| f\|,\quad f\in D, \] and \((H_ 0+V)\upharpoonright_ D\) has deficiency indices (m,m), \(m<\infty\). Then the author proves that if \(\{V_ n\}\) is a sequence of selfadjoint operators converging to V in some sense such that \(H_ 0+V\) are selfadjoint and bounded from below, then \(\{H_ 0+V_ n\}\) converges to the Friedrichs extension of \((H_ 0+V)\upharpoonright_{D(H_ 0)\cap D(V)}\) in the norm resolvent sense, even if the spectra of \(H_ 0+V_ n\) are not uniformly bounded from below. The proof relies on his result in [J. Oper. Theory 10, 209-218 (1983)].
reviewed by: Takashi Ichinose
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 3897633
zbMATH Keywords: selfadjoint operators in a Hilbert space
zbMATH Keywords: deficiency indices
zbMATH Keywords: Friedrichs extension
author: Gheorghe Nenciu
MaRDI profile type: Publication
cites work: On the one-dimensional Coulomb Hamiltonian
cites work: Essential self-adjointness of operators in ordered Hilbert space
cites work: Singular perturbation potentials
cites work: Cut-off potentials and forms extensions
cites work: Q5186190
cites work: Q5526485
description: scientific article; zbMATH DE number 3897633