Localization in \(\nu\)-dimensional incommensurate structures
type: Item
instance of: scholarly article
title: Localization in \(\nu\)-dimensional incommensurate structures
zbMATH Open document ID: 0543.35018
DOI: 10.1007/BF01211954
author: N. E. Zubov
published in: Communications in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: The authors construct a class \({\mathcal P}\) of quasi-periodic functions having singularities for which the discrete Schrödinger operator, \[ \epsilon \sum_{| e| =1}\psi(n-e)+V(x-\omega n)\psi(n)=E\quad \psi(n), \] has only pure point spectrum at small \(\epsilon\) with exponentially localized states and a dense set of eigenvalues in the real line. Here \(V\in {\mathcal P}\) and is of period 1, \(x\in R\), \(n\in {\mathbb{Z}}^{\nu}\) and \(\omega\) is a \(\nu\)-dimensional vector satisfying the diophantine condition \(\exists \gamma>0\), \(\sigma>\nu\), \(| \omega n+m| \geq \gamma | n|^{-\sigma}, \forall m\in {\mathbb{Z}}\), \(n\in {\mathbb{Z}}^{\nu}\), where \(| n| =\sum^{\nu}_{\mu =1}| n_{\mu}|\). After describing the properties of the class \({\mathcal P}\) and introducing an algebra of holomorphic kernels, the result is obtained by techniques based on the Kolmogoroff, Arnold, Moser method which was used in some previous work of the first two authors and \textit{D. Testard} [Commun. Math. Phys. 88, 207-234 (1983; Zbl 0542.35059)].
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3863679
zbMATH Keywords: discrete Schrödinger operator
zbMATH Keywords: point spectrum
zbMATH Keywords: localized states
reviewed by: Philip Schaefer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3863679