On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials
type: Item
instance of: scholarly article
title: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials
zbMATH Open document ID: 0934.34075
author: R. Deift
author: Rowan Killip
published in: Communications in Mathematical Physics
publication date: 2000-04-10T00:00:00Z
review text: Let \(H_D\), \(h_D\) respectively denote the continuous and discrete one-dimensional Schrödinger operators: \[ H_D\psi(x)= -{d^2\psi\over dx^2}+ q(x)\psi(x)\quad\text{on }L^2([0,\infty))\quad\text{with }\psi(0)= 0 \] and \[ h_D\psi(n)= \begin{cases} \psi(n+ 1)+ \psi(n- 1)+ q(n)\psi(n),\quad & n=1,2,\dots,\\ \psi(1)+ q(0)\psi(0),\quad & n= 0.\end{cases} \] It is proved that if \(q\in L^2\) then the absolutely continuous spectrum of \(H_D\) is essentially supported by \([0,\infty)\), and, if \(q\in l^2\), the absolutely continuous spectrum of \(h_D\) is essentially supported by \([-2,2]\). By essentially supported is meant that every subset of positive Lebesgue measure has positive measure with respect to some spectral measure. The results are optimal in the sense that it is known that there exist potentials decaying to zero at infinity and belonging to \(\bigcap_{2< p\leq\infty} L^p\left(\bigcap_{2< p\leq\infty}l^p\right)\) such that \(H_D(h_D)\) has purely singular continuous spectrum; [see \textit{A. Kiselev}, \textit{Y. Last} and \textit{B. Simon}, Commun. Math. Phys. 194, No. 1, 1-45 (1998; Zbl 0912.34074)]. The main ingredients of the proofs are estimates for the transmission coefficients.
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81U99
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 1334926
zbMATH Keywords: absolutely continuous spectrum
zbMATH Keywords: one-dimensional Schrödinger operators
zbMATH Keywords: transmission coefficient
reviewed by: W. D. Evans
MaRDI profile type: Publication
full work available at URL: s002200050615
OpenAlex ID: W2050242732
DOI: 10.1007/S002200050615
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description: scientific article; zbMATH DE number 1334926