 0450 Werner Kirsch, Simone Warzel
 Lifshits tails caused by anisotropic decay: the emergence of a quantumclassical regime
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Feb 27, 04

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Abstract. We investigate Lifshitstail behaviour of the integrated density of states for a wide class of Schroedinger operators with
positive random potentials. The setting includes alloytype
and Poissonian random potentials.
The considered (singlesite) 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 socalled quantum regime with Lifshits exponent $ d/2 $
if both $ \alpha_1 $ and $ \alpha_2 $ are big enough, and there is a socalled 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.
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