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 )