The spectrum of magnetic Schr\"odinger operators and $k$-form Laplacians on conformally cusp manifolds

TL;DR

This paper investigates the spectral properties of magnetic Schr"odinger operators and $k$-form Laplacians on conformally cusp manifolds, revealing conditions for the vanishing of the essential spectrum and providing eigenvalue asymptotics.

Contribution

It characterizes the essential spectrum of magnetic and $k$-form Laplacian operators on conformally cusp manifolds, including effects of magnetic fields and cohomology conditions.

Findings

Essential spectrum of Laplace operator on functions vanishes with non-integral magnetic fields.

Essential spectrum is unstable under compactly supported magnetic perturbations.

Eigenvalue asymptotics are provided for pure-point spectrum cases.

Abstract

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.