Internal Lifshits tails for random magnetic Schrödinger operators

Abstract

This paper is devoted to the study of Lifshits tails for weak random magnetic perturbations of periodic Schrödinger operators acting on L 2 ( R d ) of the form H λ , w = ( − i ∇ − λ ∑ γ ∈ Z d w γ A ( ⋅ − γ ) ) 2 + V , where V is a Z d -periodic potential, λ is positive coupling constants, ( w γ ) γ ∈ Z d are i.i.d and bounded random variables and A ∈ C 0 1 ( R d , R d ) is the single site vector magnetic potential. We prove that, for λ small, at an open band edge, a true Lifshits tail for the random magnetic Schrödinger operator occurs if a certain set of conditions on H 0 = − Δ + V and on A holds.