A spectral alternative for continuous families of self-adjoint operators
type: Item
instance of: scholarly article
title: A spectral alternative for continuous families of self-adjoint operators
zbMATH Open document ID: 1286.47010
DOI: 10.4171/JST/40
published in: Journal of Spectral Theory
publication date: 2013-05-06T00:00:00Z
review text: While eigenvalues embedded in the essential spectrum of a continuous family of self-adjoint operators in a separable Hilbert space can display very bad behaviour, it is proved in this paper that, if the set of eigenvalues is nonempty for a topologiaclly rich set of values of the parameter, then the multi-valued eigenvalue function has continuous branches. The following is proved as an application. Suppose that a one-dimensional quasi-periodic Schrödinger operator has Cantor spectrum; then a Baire generic operator in its hull does not have eigenvalues.
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 6160609
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: operator family
zbMATH Keywords: eigenvalue
zbMATH Keywords: continuous branch
zbMATH Keywords: Cantor spectrum
zbMATH Keywords: ergodic operator
author: Alexander Y. Gordon
reviewed by: Cătălin Badea
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/jst/40
OpenAlex ID: W2080355517
description: scientific article; zbMATH DE number 6160609