Integrated density of states for ergodic random Schr\"odinger operators on manifolds

TL;DR

This paper proves the existence of a non-random integrated density of states for ergodic random Schr"odinger operators on the universal cover of a compact manifold with an amenable fundamental group.

Contribution

It extends the theory of integrated density of states to Schr"odinger operators on manifolds, specifically on universal covers with amenable groups.

Findings

Existence of a non-random integrated density of states for ergodic random Schr"odinger operators on manifolds.

Application to universal covers of compact manifolds with amenable fundamental groups.

Advancement in spectral theory on geometric structures.

Abstract

We consider the Riemannian universal covering of a compact manifold $M = X / \Gamma$ and assume that $\Gamma$ is amenable. We show for an ergodic random family of Schr\"odinger operators on $X$ the existence of a (non-random) integrated density of states.