Schrödinger operators with decaying potentials: some counterexamples
type: Item
instance of: scholarly article
title: Schrödinger operators with decaying potentials: some counterexamples
zbMATH Open document ID: 1009.81019
DOI: 10.1215/S0012-7094-00-10534-0
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The Schrödinger equation is studied on the half-axis, with an asymptotically slowly decreasing potential \(V(x)\) which is asymptotically bounded by a negative power (between minus one and -- not included -- minus one half) of the coordinate. A mixed boundary condition in the origin is admitted. A review is given of the results concerning the essential spectrum (on the positive energy half-line) and its possible character (i.e., point or absolutely continuous or singularly continuous). The purpose is to construct examples which show that the results available in the literature are, in fact, optimal. An extremely well readable presentation of rigorous results (well motivated and complemented by heuristic arguments, e.g. in section 3) constructs most of its ``counterexample'' sample potentials by pasting together carefully chosen periodic pieces. Then, the bound on the embedded singular spectrum can only be shown to be optimal (in Theorem 1.3) for the asymptotic decay of \(V(x)\) between two thirds and one, which is a not entirely pleasant natural limitation of the constructive approach. Its strength, on the other hand, manifests itself at the Coulombic end (especially via sum rules of Theorem 1.5 and its Corollary 1.6). The proofs pay due attention to the effects of pasting (non-smoothness) and make ample use of periodicity of the pasted pieces, say, for related asymptotic estimates of the so called Lyapunov function (= trace of the transfer matrix), with considerations closely related to some recent Kiselev's results.
reviewed by: Miloslav Znojil
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q20
zbMATH DE Number: 1820768
zbMATH Keywords: Schrödinger operator on half-line
zbMATH Keywords: slowly decaying potentials
zbMATH Keywords: absolutely continuous spectrum
zbMATH Keywords: embedded singular spectrum
zbMATH Keywords: Hausdorff dimension of the exceptional set
zbMATH Keywords: contructive proof of optimality of its estimate
zbMATH Keywords: sum rules for asymptotically Coulomb potentials
author: Christian Remling
MaRDI profile type: Publication
Wikidata QID: Q125022616
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description: scientific article; zbMATH DE number 1820768