Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators
type: Item
instance of: scholarly article
title: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators
zbMATH Open document ID: 0931.34066
author: Y. Last
author: B. Simon
published in: Inventiones Mathematicae
publication date: 2000-02-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math-ph/9907023
full work available at URL: CaltechAUTHORS:20180320-104319293
review text: The authors give several criteria which identify the support of the absolutely continuous part of the spectrum of discrete Schrödinger operators of the form \(h : u\mapsto u(n+1)+u(n-1)+V(n)u(n)\), \(n\in {\mathbb{N}}\) or \({\mathbb{Z}}\) (with a boundary condition in the former case), in terms of bounds on the transfer matrix for \(hu=Eu\), i.e. the map \(T_E(n)\) which sends \((u(1),u(0))\) to \((u(n+1), u(n))\). A number of variants, and analogues for Schrödinger operators on \({\mathbb{R}}\), are given. \(V\) is generally assumed to be bounded. An application to almost-periodic potentials is treated. Furthermore, an argument by \textit{D. Ruelle} [Publ. Math., Inst. Hautes Etud. Sci. 50, 275-306 (1979)] is used to find a sufficient condition on \(\|T_E(n)\|\) which ensures the existence of a square integrable eigenfunction. For applications to sparse potentials, random potentials, and \(n^{-\alpha}\) potentials, the authors refer to a companion paper with \textit{A. Kiselev} [Commun. Math. Phys. 194, No. 1, 1-45 (1998; Zbl 0912.34074)].
reviewed by: Satyanad Kichenassamy
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 47B39
zbMATH DE Number: 1275763
zbMATH Keywords: discrete and continuous one-dimensional Schrödinger operators
zbMATH Keywords: absolutely continuous spectrum
zbMATH Keywords: almost-periodic potentials
MaRDI profile type: Publication
Wikidata QID: Q56688300
OpenAlex ID: W2076700191
arXiv ID: math-ph/9907023
DOI: 10.1007/S002220050288
description: scientific article