We consider non-ergodic magnetic random Schrödinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of arguments from Klein (Commun Math Phys 323(3):1229–1246, 2013), combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from Borisov et al. (J Math Phys, arXiv:1512.06347 [math.AP]). 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 $$E_0(\infty ) \in (0, \infty ]$$ E 0 ( ∞ ) ∈ ( 0 , ∞ ] , 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 without magnetic field.
Annales Henri Poincaré – Springer Journals
Published: Dec 11, 2017
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