## 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

### MSC:

 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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### References:

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