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

Abstract

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: \mathbbm{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.

Revised and generalized version