On the essential selfadjointness of stochastic Schrödinger operators
type: Item
instance of: scholarly article
title: On the essential selfadjointness of stochastic Schrödinger operators
zbMATH Open document ID: 0543.60069
DOI: 10.1215/S0012-7094-83-05055-X
author: Fabio Martinelli
author: Werner Kirsch
published in: Duke Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Let \(V_{\omega}\) be a random field on \({\mathbb{R}}^ d\) and consider the random Schrödinger operator \(H_{\omega}=-\Delta +V_{\omega}\). Let furthermore \(C_ y\) be the unit cube around the point y. The main result of the paper states that \(H_{\omega}\) is essentially selfadjoint on \(C_ 0^{\infty}({\mathbb{R}}^ d)\) if \(E(|(\int_{C_ y}| V_{\omega}(x)|^ qdx)^{1/q}|^ r)\leq C<\infty\) for \(q>q_ 0(d)\) and \(r>r_ 0(d,q)\) (E denotes the expectation with respect to \(\omega)\). Moreover it is clear from the proof that adding a potential \(W(x)\geq -cx^ 2\)-d does not disturb the essential selfadjointness \(q_ 0(d)=\max(2,d/2).\) There is some confusion about \(r_ 0(d,q)\) in the paper under review. The correct \(r_ 0\) is given by \(r_ 0(d,q)=dq/2(q-p)\) which should replace (3) in the paper. In the last section applications are discussed which cover many random potentials of physical and mathematical interest.
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 3864234
zbMATH Keywords: random Schrödinger operator
zbMATH Keywords: essentially selfadjoint
zbMATH Keywords: random potentials
MaRDI profile type: Publication
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full work available at URL: s0012-7094-83-05055-x
OpenAlex ID: W1495148021
description: scientific article; zbMATH DE number 3864234