Rank one perturbations at infinite coupling
type: Item
instance of: scholarly article
title: Rank one perturbations at infinite coupling
zbMATH Open document ID: 0828.47009
author: B. Simon
author: Fritz Gesztesy
published in: Journal of Functional Analysis
publication date: 1996-01-08T00:00:00Z
review text: The authors discuss rank one perturbations \(A_\alpha = A + \alpha (\varphi, \cdot) \varphi\), \(\alpha \in \mathbb{R}\), \(A \geq 0\) self-adjoint. Let \(d \mu_\alpha(x)\) be the spectral measure defined by \((\varphi, (A_\alpha - z)^{-1} \varphi) = \int d\mu_\alpha(x)/(x-z)\). We prove there is a measure \(d\rho_\infty\) which is the weak limit of \((1 + \alpha^2) d\mu_\alpha(x)\) as \(\alpha \to \infty\). If \(\varphi\) is cyclic for \(A\), then \(A_\infty\), the strong resolvent limit of \(A_\alpha\), is unitarily equivalent to multiplication by \(x\) on \(L^2 (\mathbb{R}, d \rho_\infty)\). This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case.
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 764007
zbMATH Keywords: infinite coupling
zbMATH Keywords: rank one perturbations
zbMATH Keywords: spectral measure
zbMATH Keywords: unitarily equivalent to multiplication
zbMATH Keywords: Sturm-Liouville operators on half-lines
MaRDI profile type: Publication
full work available at URL: jfan.1995.1030
OpenAlex ID: W2025143162
DOI: 10.1006/JFAN.1995.1030
description: scientific article; zbMATH DE number 764007