Examples of discrete Schrödinger operators with pure point spectrum
type: Item
instance of: scholarly article
title: Examples of discrete Schrödinger operators with pure point spectrum
zbMATH Open document ID: 0532.35070
DOI: 10.1007/BF01211953
author: Jürgen Pöschel
published in: Communications in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: The discrete Schrödinger operator \((Hu)_ i=\epsilon \sum_{| j| =1}u_{i+j}+q_ iu_ i, i\in Z^ m\) on \(\ell^ 2(Z^ m)\), \(m\geq 1\), with small coupling \(\epsilon\) is considered. Examples of limit periodic potentials \((q_ i)\) are constructed, for which H has pure point spectrum with a complete set of exponentially localized eigenstates. The spectrum can be an interval or a Cantor set. H is considered as a perturbation of the multiplicative operator \((Qu)_ i=q_ iu_ i\). If Q satisfies certain small divisor conditions, then another diagonal operator \(\hat Q\) of order \(\epsilon^ 2\) and an isomorphism V of \(\ell^ 2(Z^ m)\) via KAM-iteration are constructed such that \(V^{-1}(H+\hat Q)V=Q\). Thus, actually an inverse spectral problem is solved. These results are special cases of more general results about perturbations of multiplication operators. They include also a number of other known examples of discrete Schrödinger operators with pure point spectrum.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 81Q15
zbMATH DE Number: 3844264
zbMATH Keywords: discrete Schrödinger operators
zbMATH Keywords: pure point spectrum
zbMATH Keywords: perturbations of multiplication operators
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01211953
OpenAlex ID: W2036441968
description: scientific article; zbMATH DE number 3844264