Mathematical Physics, Analysis and Geometry

, Volume 8, Issue 3, pp 257–285 | Cite as

Lifshits Tails Caused by Anisotropic Decay: The Emergence of a Quantum-Classical Regime

  • Werner KirschEmail author
  • Simone Warzel


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: ℝd→[0,∞[ 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 |(x1,x2)|→∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α1 and α2 are big enough, and there is a so-called classical regime with Lifshits exponent depending on α1 and α2 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 α1 and α2 simultaneously.


random Schrödinger operators integrated density of states Lifshits tails 


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© Springer 2005

Authors and Affiliations

  1. 1.Institut für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Institut für Theoretische PhysikUniversität Erlangen–NürnbergErlangenGermany

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