The fate of Lifshits tails in magnetic fields. (English) Zbl 1081.82572
Summary: 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 nonnegative, algebraically decaying, single-impurity potential we prove that the leading asymptotic behavior 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 eingenspace of any Landau level exhibits the same behavior. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.
MSC:
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
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