Wegner estimate for random divergence-type operators monotone in the randomness

TL;DR

This paper establishes a Wegner estimate for a broad class of random divergence-type operators that are monotone in randomness, utilizing unique continuation estimates and eigenvalue lifting techniques.

Contribution

It introduces a Wegner estimate for general monotone divergence-type operators with complex, possibly non-linear, random perturbations.

Findings

Proves Wegner estimate for a wide class of random operators

Utilizes unique continuation estimates for the gradient

Handles non-linear dependence on random parameters

Abstract

In this note, a Wegner estimate for random divergence-type operators that are monotone in the randomness is proven. The proof is based on a recently shown unique continuation estimate for the gradient and the ensuing eigenvalue liftings. The random model which is studied here contains quite general random perturbations, among others, some that have a non-linear dependence on the random parameters.