The spectrum of operators elliptic along the orbits of \({\mathbb{R}}^ n\) actions. (English) Zbl 0643.58033

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.
Reviewer: M.K.Murray


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J22 Exotic index theories on manifolds
35P99 Spectral theory and eigenvalue problems for partial differential equations
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