Endpoints of the spectrum of periodic operators are generically simple.
type: Item
instance of: scholarly article
title: Endpoints of the spectrum of periodic operators are generically simple.
zbMATH Open document ID: 1040.35050
author: Frédéric Klopp
author: J. V. Ralston
published in: Methods and Applications of Analysis
publication date: 2002-06-19T00:00:00Z
review text: The paper deals with the Schrödinger operator in \({\mathbb R}^n\), \[ H(V)=-\Delta+V, \] where the potential \(V\) is a real measurable \({\mathbb Z}^n\)-periodic bounded function. The spectrum of \(H(V)\) is a union of bands \[ B_j=\{\lambda_j(k,V)\mid k\in{\mathbb T}^n\}, \] where \(\lambda_j(k,V)\) are the eigenvalues (counted with multiplicities, and in increasing order) of the reduced operators \[ H(k,V)=-(\partial+2\pi\text{i}k)^2+V \] in \(L^2({\mathbb T}^n)\). A point \(\lambda\) in the spectrum of \(H(V)\) is called simple if there exists a unique \(j\geq1\) such that \(\lambda_j(k,V)=\lambda\) for some \(k\in{\mathbb T}^n\). Let \({\mathcal V}_l\) be the set of bounded \({\mathbb Z}^n\)-periodic potentials \(V\) for which \(H(V)\) has at least \(l\) open gaps. Then, the main result of the paper under review shows that the set of potentials \(V\) for which the endpoints of at least \(l\) gaps in the spectrum of \(H(V)\) are simple is a dense open subset of \({\mathcal V}_l\) with respect to the \(L^\infty({\mathbb T}^n)\)-topology.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 1757312
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: periodic potential
zbMATH Keywords: band
zbMATH Keywords: Floquet eigenvalue
zbMATH Keywords: finite gaps
reviewed by: Daniel Beltiţă
MaRDI profile type: Publication
full work available at URL: maa.2000.v7.n3.a2
OpenAlex ID: W1588949043
DOI: 10.4310/MAA.2000.V7.N3.A2
description: scientific article; zbMATH DE number 1757312