04-50 Werner Kirsch, Simone Warzel
Lifshits tails caused by anisotropic decay: the emergence of a quantum-classical regime (361K, ps) Feb 27, 04
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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.

Files: 04-50.src( 04-50.keywords , LifInh.ps )