The spectrum of operators elliptic along the orbits of \({\mathbb{R}}^ n\) actions
type: Item
instance of: scholarly article
title: The spectrum of operators elliptic along the orbits of \({\mathbb{R}}^ n\) actions
zbMATH Open document ID: 0643.58033
DOI: 10.1007/BF01212421
author: Jerome Kaminker
author: Jingbo Xia
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: A method of studying differential operators on an open manifold, developed by A. Connes, is to consider the open manifold as a leaf in a foliation of a compact manifold. The authors apply this method in the more general setting of a group action of \(R^ n\) on a compact Hausdorff space. They extend the ideas and methods of Bellisard and Testard to prove a ``spectral duality theorem'' in this general setting. The main application is a proof that a periodic elliptic operator on \(R^ n\) has no eigenvalues, off of the discontinuities of the spectral density function, which have \(L^ 2\)-eigenfunctions. The authors hope to use the general setting of their results for future applications to almost-periodic Schrödinger operators.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 35P99
zbMATH DE Number: 4048720
zbMATH Keywords: foliation
zbMATH Keywords: group action
zbMATH Keywords: operator algebra
zbMATH Keywords: spectral duality
zbMATH Keywords: periodic elliptic operator
zbMATH Keywords: almost-periodic Schrödinger operators
reviewed by: Michael K. Murray
MaRDI profile type: Publication
cites work: Q5607725
cites work: Q4200733
cites work: The rotation number for almost periodic potentials
cites work: ON THE COINCIDENCE OF THE SPECTRA OF RANDOM ELLIPTIC OPERATORS
cites work: Spectral invariants of foliations
cites work: Q5703967
cites work: Almost periodic Schrödinger operators: A review
cites work: THE SPECTRAL THEORY AND THE INDEX OF ELLIPTIC OPERATORS WITH ALMOST PERIODIC COEFFICIENTS
cites work: Spectrum and its distribution function for a transversally elliptic operator
cites work: Q4751318
description: scientific article; zbMATH DE number 4048720