Integrated density of states for random metrics on manifolds

TL;DR

This paper investigates the spectral properties of ergodic random Schrödinger operators on manifolds with random metrics and potentials, establishing key results like measurability, spectral invariance, and the existence of an integrated density of states.

Contribution

It introduces a framework for analyzing random Schrödinger operators with random metrics, proving measurability, spectral invariance, and deriving a trace formula, which are novel in this context.

Findings

Proved measurability of the random operators

Established almost sure spectral invariance

Demonstrated existence of a self-averaging integrated density of states

Abstract

We study ergodic random Schr"odinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a selfaveraging integrated density of states and a \v{S}ubin type trace formula.