The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds

Details The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds

2005-07-21

arxiv.org/abs/math/0507443v1

Mathematics

Differential Geometry

32 pages, no figures

Scientific paper

We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an integral relative cohomology class. It follows that the essential spectrum is not stable by perturbation even by a compactly supported magnetic field. We also treat magnetic operators perturbed with electric fields. In the same context we describe the essential spectrum of the $k$-form Laplacian. This is shown to vanish precisely when the $k$ and $k-1$ de Rham cohomology groups of the boundary vanish. In all the cases when we have pure-point spectrum we give Weyl-type asymptotics for the eigenvalue-counting function. In the other cases we describe the essential spectrum.

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