The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials
type: Item
instance of: scholarly article
title: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials
zbMATH Open document ID: 0892.34075
author: Michael Christ
author: Alexander Kiselev
author: Christian Remling
published in: Mathematical Research Letters
publication date: 1998-07-05T00:00:00Z
review text: The authors consider the spectrum of the self-adjoint one-dimensional Schrödinger operator \(H = -d^2/dx^2 + V(x)\), acting on the Hilbert space \(L_2(0, \infty)\), with a boundary condition at \(x=0\). Considering the potential \(V\) as a perturbation, the authors present three theorems providing the conditions when suitable smallness assumptions on \(V\) guarantee stability of the absolutely continuous spectrum of \(H_0 = -d^2/dx^2\). In particular, the authors present a sharper version of a theorem of \textit{A. Kiselev} [Commun. Math. Phys. 179, No. 2, 377-400 (1996; Zbl 0858.34071)], stating that the absolutely continuous spectrum is preserved if \(V(x) = O(x^{-\alpha})\) \((\alpha >2/3)\).
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 1117493
zbMATH Keywords: continuous spectrum
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: spectral problems
reviewed by: Yuri S. Kivshar
MaRDI profile type: Publication
full work available at URL: mrl.1997.v4.n5.a9
OpenAlex ID: W3021306681
DOI: 10.4310/MRL.1997.V4.N5.A9
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description: scientific article; zbMATH DE number 1117493