Wegner estimate and disorder dependence for alloy-type Hamiltonians with bounded magnetic potential

TL;DR

This paper establishes an optimal Wegner estimate for non-ergodic magnetic random Schrödinger operators with bounded magnetic potentials, analyzing how disorder influences the estimate's constants across energy levels.

Contribution

It extends Klein's Wegner estimate to operators with bounded magnetic vector potentials and examines the disorder dependence of the Wegner constant.

Findings

Optimal Wegner estimate valid at all energies.

Wegner constant cannot tend to zero with increasing disorder above a certain energy.

Generalization of previous results to magnetic operators.

Abstract

We consider non-ergodic magnetic random Sch\"odinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of the arguments from [Kle13], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [BTV15]. This generalizes Klein's result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold $E_0(\infty) \in (0, \infty]$, it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian with vanishing magnetic field.