Landau Hamiltonians with Unbounded Random Potentials

Abstract

We prove the almost sure existence of a 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 is assumed to be absolutely continuous. The corresponding densityg has support equal to $\mathbb{R}$, and satisfies$$, for some ∈ > 0. This includes the case of Gaussian distributions. We show that the almost sure spectrum ∑ is $\mathbb{R}$, provided the magnetic field B≠0. We prove that for each positive integer n, there exists a field strength B _n , such that for all B>B _n , the almost sure spectrum ∑ is pure point at all energies $E \leqslant (2n + 3)B - \mathcal{O}(B^{ - 1} )$ except in intervals of width$\mathcal{O}(B^{ - 1} )$ about each lower Landau level $E_m (B) \equiv (2m + 1)B$ , for m < n. We also prove that for any B≠0, the integrated density of states is Lipschitz continuous away from the Landau energiesE _n (B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.