The spectrum of Schr\"odinger operators and Hodge Laplacians on conformally cusp manifolds

TL;DR

This paper analyzes the spectrum of Laplacians on conformally cusp manifolds, revealing conditions for essential spectrum vanishing, providing eigenvalue asymptotics, and exploring Schr"odinger operators with unbounded potentials.

Contribution

It offers a detailed spectral analysis of Laplacians on conformally cusp manifolds, including criteria for essential spectrum and eigenvalue asymptotics, and corrects previous literature on hyperbolic manifolds.

Findings

Essential spectrum vanishes when boundary cohomology groups vanish.

Weyl-type asymptotics are established for discrete eigenvalues.

Certain Schr"odinger operators with unbounded potentials have compact resolvent.

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 Schr\"odinger operator has compact resolvent, although $V$ tends to $-\infty$ in most of the infinity. 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 four whose cusps are rational homology spheres.