The Fate of Lifshitz Tails in Magnetic Fields

Abstract

We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.

arXiv comment19 pages LaTeX

References in corpus (2)

• The Non-Commutative Geometry of the Quantum Hall Effect
• Landau Hamiltonians with Random Potentials: Localization and the Density of States

Cited by in corpus (11)

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• Recent developments in quantum mechanics with magnetic fields
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• Classical magnetic Lifshits tails in three space dimensions: impurity potentials with slow anisotropic decay
• Scattering for Schrödinger operators with potentials concentrated near a subspace
• Surface Lifshits tails for random quantum Hamiltonians