Propagation properties for Schrödinger operators affiliated with certain \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Propagation properties for Schrödinger operators affiliated with certain \(C^*\)-algebras
zbMATH Open document ID: 1029.81066
published in: Annales Henri Poincaré
publication date: 2003-03-20T00:00:00Z
review text: The paper under review is concerned with propagation properties of scattering states of selfadjoint \(n\)-dimensional Schrödinger operators \(H=-\Delta +V\) with potential \(V\). Remember that the scattering states of \(H\) are defined by the property that they propagate away from each bounded region of the configuration space \({\mathbb R}^n\) as time \(t\to\pm\infty\). If the potential \(V\) tends to zero sufficiently rapidly at infinity, a description of the behaviour of \(\text{ e}^{-\text{ i}tH}f\) for a scattering state \(f\) at large times \(t\) is provided by standard scattering theory. On the other hand, in more complicated situations (e.g., if the asymptotic behaviour of \(V\) is anisotropic), little is known about the propagation of scattering states, and it is expected that certain asymptotic regions of configuration space should be inaccessible to states of certain energies. Using a careful spectral analysis of the Hamiltonians \(H\) under study, detailed results about the propagation and non-propagation of states for one-dimensional Schrödinger operators and multi-dimensional operators periodic in all but one dimension are obtained in [\textit{E. B. Davies} and \textit{B. Simon}, Commun. Math. Phys. 63, 277-301 (1978; Zbl 0393.34015)]. In the paper under review, a different method for obtaining non-propagation properties, based on a recent approach to spectral theory in the framework of \(C^*\)-algebras, is proposed. Along these lines, an abstract theorem on non-propagation and applications to Schrödinger Hamiltonians that are asymptotically periodic in several cones are given.
reviewed by: Gerald Hofmann
Mathematics Subject Classification ID: 81U05
Mathematics Subject Classification ID: 81R15
Mathematics Subject Classification ID: 46L06
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 1874656
zbMATH Keywords: scattering theory
zbMATH Keywords: anisotropic Schrödinger operators
zbMATH Keywords: C*-techniques
author: Werner O. Amrein
author: Marius Măntoiu
author: Radu Purice
MaRDI profile type: Publication
full work available at URL: s000230200003
OpenAlex ID: W2052694151
DOI: 10.1007/S000230200003
description: scientific article; zbMATH DE number 1874656