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

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 $E_0(\mathrm{\infty })\in (0,\mathrm{\infty }]$, 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.