Surface Lifshits tails for random quantum Hamiltonians

Abstract

We consider Schrödinger operators on L2(ℝd)⊗L2(ℝℓ) of the form Hω=H⊥⊗I∥+I⊥⊗H∥+Vω, where H⊥ and H∥ are Schrödinger operators on L2(ℝd) and L2(ℝℓ), respectively, and Vω(x,y):=∑ξ∈ℤdλξ(ω)v(x−ξ,y),x∈ℝd,y∈ℝℓ is a random “surface potential.” We investigate the behavior of the integrated density of surface states of Hω near the bottom of the spectrum and near internal band edges. The main result of the current paper is that, under suitable assumptions, the behavior of the integrated density of surface states of Hω can be read off from the integrated density of states of a reduced Hamiltonian H⊥+Wω where Wω is a quantum mechanical average of Vω with respect to y∈ℝℓ. We are particularly interested in cases when H⊥ is a magnetic Schrödinger operator, but we also recover some of the results from Kirsch and Warzel [J. Funct. Anal. 230, 222–250 (2006)] for non-magnetic H⊥.