LANDAU HAMILTONIANS WITH UNBOUNDED RANDOM POTENTIALS

We prove the almost sure existence of pure point spectrum for the two- dimensional Landau Hamiltonian with an unbounded Anderson-like random potential, provided that the magnetic field is sufficiently large. For these models, the probability distribution of the coupling constant i s assumed to be absolutely continuous. The corresponding density g has support equal to R, and satisfies sup(lambda is an element of R){lamb da(3+epsilon)g(lambda)} < infinity, for some epsilon > 0. This include s the case of Gaussian distributions. We show that the almost sure spe ctrum Sigma is R, provided the magnetic field B not equal 0. We prove that for each positive integer n, there exists a field strength B-n, s uch that for all B > B-n, the almost sure spectrum Sigma is pure point at all energies E less than or equal to (2n + 3)B - O(B-1) except in intervals of width O(B-1) about each lower Landau level E-m(B) = (2m 1)B, for m < n. We also prove that for any B not equal 0, the integra ted density of states is Lipschitz continuous away from the Landau ene rgies E-n(B). This follows from a new Wegner estimate for the finite-a rea magnetic Hamiltonians with random potentials.