Stability criteria for the Weyl \(m\)-function
type: Item
instance of: scholarly article
title: Stability criteria for the Weyl \(m\)-function
zbMATH Open document ID: 0919.34075
DOI: 10.1023/A:1009759202730
author: D. B. Pearson
author: Werner O. Amrein
published in: Mathematical Physics, Analysis and Geometry
publication date: 1999-08-25T00:00:00Z
review text: The paper presents a new approach to the spectral theory of the Schrödinger operator on the half-line, based on the analysis of Weyl-Titchmarsh \(m\)-function and its boundary values. The investigation is based on the Riccati equation: \({d\over dx}h(x,\lambda)= V(x)-\lambda- (h(x,\lambda))^2\), corresponding to the Schrödinger operator: \(-{d^2\over dx^2}+V(x)\). \(\lambda\) is the (real) spectral parameter. The authors show large \(x\) asymptotics of families of solutions to the Riccati equation, as \(\lambda\) is varied, imply explicit bounds of \(m_+(\lambda)\), the boundary value of the \(m\)-function (or of \(m_\alpha\) if Neumann or other boundary conditions are imposed). A family of solutions exhibiting a so-called \(\delta\)-clustering property is shown to imply precise estimates for the complex boundary values of the Weyl \(m\)-function and the spectral measure, and leads to an analysis of the absolutely continuous component of the spectral measure in terms of stability criteria for the Riccati equation.
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 1247684
zbMATH Keywords: spectral measure
zbMATH Keywords: \(\delta\)-clustering
zbMATH Keywords: Weyl-Titchmarsh \(m\)-function
MaRDI profile type: Publication
full work available at URL: a:1009759202730
OpenAlex ID: W115241461
description: scientific article; zbMATH DE number 1247684