Lifshitz tail for continuous Anderson models driven by Lévy operators

Abstract

We investigate the behavior near zero of the integrated density of states for random Schrödinger operators [Formula: see text] in [Formula: see text], [Formula: see text], where [Formula: see text] is a complete Bernstein function such that for some [Formula: see text], one has [Formula: see text], [Formula: see text], and [Formula: see text] is a random nonnegative alloy-type potential with compactly supported single site potential [Formula: see text]. We prove that there are constants [Formula: see text] such that [Formula: see text] where [Formula: see text] is the common cumulative distribution function of the lattice random variables [Formula: see text]. For typical examples of [Formula: see text] the constants [Formula: see text] and [Formula: see text] can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, the large class of operator monotone functions of the Laplacian. This class includes both local and nonlocal kinetic terms such as the Laplace operator, its fractional powers, the quasi-relativistic Hamiltonians and many others.