The modulus of continuity of Wegner estimates for random Schr\"odinger operators on metric graphs

TL;DR

This paper establishes a Wegner estimate for random Schr"odinger operators on infinite metric graphs with alloy type potentials, showing the estimate's dependence on the single site distribution and its independence from energy.

Contribution

It introduces a Wegner estimate for metric graph Schr"odinger operators that reflects the single site distribution's modulus of continuity, independent of energy levels.

Findings

Wegner estimate reproduces the modulus of continuity of the single site measure

Constant in the estimate is independent of energy

Applicable to finite volume subgraphs of infinite metric graphs

Abstract

We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schr\"odingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of continuity of the single site distribution measure. The Wegner constant is independent of the energy.