Lifshits tails caused by anisotropic decay: the emergence of a quantum-classical regime

We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f : Rd → [0,∞[ decay at infinity in an anisotropic way, for example, f(x1, x2) ∼ (|x1|α1 +|x2|α2 )−1 as |(x1, x2)| → ∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α1 and α2 are big enough, and there is a so-called classical regime with Lifshits exponent depending on α1 and α2 if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on α1 and α2 simultaneously. Dedicated to the memory of G. A. Mezincescu ( 1943 – 2001 ). CONTENTS