The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds

Golénia, Sylvain; Moroianu, Sergiu

doi:10.1090/S0002-9947-2011-05216-5

The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds
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by Sylvain Golénia and Sergiu Moroianu PDF

Trans. Amer. Math. Soc. 364 (2012), 1-29 Request permission

Abstract:

We describe the spectrum of the $k$-form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the $k$ and $k-1$ de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials $V$ such that the Schrödinger operator has compact resolvent, although in most directions the potential $V$ tends to $-\infty$. We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension 4 whose cusps are rational homology spheres.

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