Werner Kirsch, Simone Warzel
Lifshits tails caused by anisotropic decay: the emergence of a quantum-classical regime
(361K, ps)
ABSTRACT. We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schroedinger operators with
positive random potentials. The setting includes alloy-type
and Poissonian random potentials.
The considered (single-site) impurity
potentials $ f: R^d \to [0, \infty[ $ decay at infinity in an anisotropic way, for example,
$ f(x_1,x_2)\sim (|x_1|^{\alpha_1}+|x_2|^{\alpha_2})^{-1} $ as
$ |(x_1,x_2)| \to \infty $.
As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent $ d/2 $
if both $ \alpha_1 $ and $ \alpha_2 $ are big enough, and there is a so-called classical regime with Lifshits exponent depending on
$ \alpha_1 $ and $\alpha_2$ 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 $ \alpha_1 $ and $ \alpha_2 $ simultaneously.