On inverse spectral theory for selfadjoint extensions: Mixed types of spectra
type: Item
instance of: scholarly article
title: On inverse spectral theory for selfadjoint extensions: Mixed types of spectra
zbMATH Open document ID: 0997.47005
author: S. Albeverio
author: Hagen Neidhardt
author: J. Brasche
published in: Journal of Functional Analysis
publication date: 2002-09-17T00:00:00Z
review text: Let \(H\) be a symmetric operator in a separable Hilbert space \({\mathcal H}\). Suppose that \(H\) has some gap \(J=(a,b), -\infty\leq a < b < \infty\), i.e \[ (Hf,f) \geq b\|f\|^2, f\in{\mathcal D}(H), \text{ if} \;a=-\infty, \] \[ \|(H-{{a+b}\over 2})f\|\geq {{b-a}\over 2}\|f\|, f\in{\mathcal D}(H), \text{ if} \;\infty < a. \] The authors investigate spectral properties of selfadjoint extensions of \(H\) inside \(J\). Under some weak assumptions about the operator \(H\) which are satisfied. e.g.. provided the deficiency indices of \(H\) are infinite and the operator \((H-\lambda)^{-1} \) is compact for one regular point \(\lambda\) of \(H\), it is proved that for every (auxiliary) selfadjoint operator \(M'\) in the Hilbert space \(\mathcal H\) and every open subset \(J_0\) of the gap \(J\) there exists a selfadjoint extension \(\hat H\) of \(H\) such that inside \(J\) the operator \(\hat H\) has the same absolutely continuous and the same point spectrum as the given operator \(M'\) and the singular continuous spectrum of \(\hat H\) on \(J\) equals the closure of \(J_0\) in \(J\). Moreover a method of construction of such a selfadjoint extension \(\hat H\) is proposed. This method is applied to the construction of new kinds of selfadjoint realizations of the Laplacian on a bounded domain \(\Omega\) in \({\mathbf R}^d, d > 1\) with spectral properties very different from the spectral properties of the selfadjoint realizations known before.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 1213982
zbMATH Keywords: selfadjoint extensions
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: selfadjoint realizations of the Laplacian on a bounded domain
MaRDI profile type: Publication
full work available at URL: jfan.1997.3190
OpenAlex ID: W2091950597
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DOI: 10.1006/JFAN.1997.3190
description: scientific article; zbMATH DE number 1213982