Cantor spectrum for the almost Mathieu equation
type: Item
instance of: scholarly article
title: Cantor spectrum for the almost Mathieu equation
zbMATH Open document ID: 0516.47018
DOI: 10.1016/0022-1236(82)90094-5
author: B. Simon
author: Jean Bellissard
published in: Journal of Functional Analysis
publication date: 1982-01-01T00:00:00Z
review text: For a dense \(G_\delta\) of pairs \((\lambda, \alpha)\) in \(\mathbb R^2\), the authors prove that the operator \[ (Hu)(n)=u(n+1)+u(n-1)+\lambda\,\cos(2\pi\alpha n+\theta)u(n) \] has a nowhere dense spectrum. Additional results are proved for rational \(\alpha\). For example, if \(\alpha=(2p+1)/2q\) and \(\theta=0\) [resp. \(\pi/2q\)] if \(q\) is even [resp. odd] then one spectral gap is closed for all \(\lambda\). The paper concludes with a conjecture about how the main theorem might be strengthened using \(K\)-theory or homotopy.
reviewed by: Alan L. Andrew
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 3816429
zbMATH Keywords: spectral gap
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: homotopy
Wikidata QID: Q56688303
MaRDI profile type: Publication
full work available at URL: 0022-1236(82)90094-5
OpenAlex ID: W2068177641
cites work: The asymptotics of the gap in the Mathieu equation
cites work: Almost periodic Schrödinger operators. I: Limit periodic potentials
cites work: NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES
cites work: Double wells
cites work: The recurrent Hill's equation
cites work: The rotation number for almost periodic potentials
cites work: An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum
cites work: Q3039940
cites work: Q3665727
description: scientific article; zbMATH DE number 3816429