Lifschitz tail for alloy-type models driven by the fractional Laplacian

Abstract

We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators (−Δ)α/2+Vω in L2(Rd) for the full range of α∈(0,2] and a fairly large class of random nonnegative alloy-type potentials Vω. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limitlimλ→0⁡λd/αln⁡N(λ)=−Cωd(λd(α))d/α, with C∈(0,∞]. The constant C is finite if and only if the common distribution of the lattice random variables charges {0}. In this case, the constant C is expressed explicitly in terms of this distribution. In the limit formula, λd(α) denotes the Dirichlet ground-state eigenvalue of the operator (−Δ)α/2 in the unit ball in Rd, and ωd is the volume of this ball.