Jm. Combes et Pd. Hislop, LANDAU HAMILTONIANS WITH RANDOM POTENTIALS - LOCALIZATION AND THE DENSITY-OF-STATES, Communications in Mathematical Physics, 177(3), 1996, pp. 603-629
We prove the existence of localized states at the edges of the bands f
or the two-dimensional Landau Hamiltonian with a random potential, of
arbitrary disorder, provided that the magnetic field is sufficiently l
arge. The corresponding eigenfunctions decay exponentially with the ma
gnetic field and distance. We also prove that the integrated density o
f states is Lipschitz continuous away from the Landau energies. The pr
oof relies on a Wegner estimate for the finite-area magnetic Hamiltoni
ans with random potentials and exponential decay estimates for the fin
ite-area Green's functions. The proof of the decay estimates for the G
reen's functions uses fundamental results from two-dimensional bond pe
rcolation theory.