J.M. Combes, P.D. Hislop
Landau Hamiltonians with Random Potentials: Localization and the Density of States
The paper is published: Commun. Math. Phys. 177, 3 (1996) 603-629

MSC:

60K40 Other physical applications of random processes

82C44 Dynamics of disordered systems (random Ising systems, etc.)

Abstract: We prove the existence of localized states at the edges of the
bands for the two-dimensional Landau Hamiltonian with a random potential,
of arbitrary disorder, provided that the magnetic field is
sufficiently large. The corresponding eigenfunctions decay
exponentially with the magnetic field and distance. We also prove that the
integrated density
of states is Lipschitz continuous away from the Landau energies.
The proof relies on a Wegner estimate for the finite-area magnetic
Hamiltonians with random potentials and exponential decay estimates
for the finite-area Green's functions. The proof of the decay estimates
for the Green's functions uses
fundamental results from two-dimensional bond percolation theory.

Keywords: Landau Hamiltonians, random operators, localization