Some abstract Wegner estimates with applications

TL;DR

This paper establishes abstract Wegner estimates for random self-adjoint operators, providing new bounds applicable to various models including Anderson Hamiltonians and quantum graphs, even with correlated and general distributions of randomness.

Contribution

It introduces a unified abstract framework for Wegner estimates applicable to diverse models with correlated and general random parameters.

Findings

Wegner bounds for discrete and continuous Anderson Hamiltonians

Wegner bounds for quantum graphs with random parameters

Applicable to correlated and general distribution randomness

Abstract

We prove some abstract Wegner bounds for random self-adjoint operators. Applications include elementary proofs of Wegner estimates for discrete and continuous Anderson Hamiltonians with possibly sparse potentials, as well as Wegner bounds for quantum graphs with random edge length or random vertex coupling. We allow the coupling constants describing the randomness to be correlated and to have quite general distributions.