LANDAU HAMILTONIANS WITH RANDOM POTENTIALS - LOCALIZATION AND THE DENSITY-OF-STATES

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.