On energy gaps in a new type of analytically solvable model in quantum mechanics
type: Item
instance of: scholarly article
title: On energy gaps in a new type of analytically solvable model in quantum mechanics
zbMATH Open document ID: 0669.47002
DOI: 10.1016/0022-247X(88)90003-0
author: Werner Kirsch
author: Fritz Gesztesy
author: Helge Holden
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The authors study spectral properties of one-dimensional Schrödinger Hamiltonians of the form \[ (*)\quad H_{\{\beta_ j\}}=-d^ 2/dx^ 2+\sum_{j\in {\mathbb{Z}}} \beta_ jV_ j \] where \(V_ j\) is Dirac's delta function/derivative of Dirac's delta function at a point \(y_ j\in {\mathbb{R}}\) for all \(j\in {\mathbb{Z}}.\) In particular various generalizations of the Saxon-Hutner conjecture are proved: Let \(\{y_ j\}=a{\mathbb{Z}}\), and let \(H_{\beta}\) denote the operator (*) with \(\beta_ j=\beta\), \(j\in {\mathbb{Z}}\). Then it is e.g. proved that if (a,b)\(\subset \cap_{\beta \in \{\beta_ j\}} \rho (H_{\beta})\) then \((a,b)\subset \rho (H_{\{\beta_ j\}})\) where \(\rho\) (H) denotes the resolvent set of an operator H. Finally the authors analyze the case when the \(\beta_ j\) are random variables. Assume the \(\beta_ j's\) are independent identically distributed random variables with support in [0,\(\infty)\). Let \(\gamma\) denote the inf of the distribution of \(\beta_ j.\) Then \(\Sigma =\sigma (H_{\gamma})=\cup_{m\in {\mathbb{Z}}} [a_ m,b_ m]\) where \(\Sigma =\alpha (H_{\{\beta_ j\}})\) almost surely. For more extensive treatment of these operators we refer to the book of \textit{S. Albeverio}, \textit{F. Gesztesy}, R. Høegh-Krohn, \textit{H. Holden} [Solvable Models in Quantum Mechanics, Springer-Verlag (1988)].
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 4096178
zbMATH Keywords: spectral properties of one-dimensional Schrödinger Hamiltonians
zbMATH Keywords: Saxon- Hutner conjecture
zbMATH Keywords: independent identically distributed random variables
MaRDI profile type: Publication
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full work available at URL: 0022-247x(88)90003-0
OpenAlex ID: W2033828368
description: scientific article; zbMATH DE number 4096178