## Abstract

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 is assumed to be absolutely continuous. The corresponding density g has support equal to ℝ, and satisfies sup_{λ∈ℝ}{λ^{3+∈}g(λ)} < ∞, for some ∈ > 0. This includes the case of Gaussian distributions. We show that the almost sure spectrum Σ is 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 Σ is pure point at all energies E ≤ (2n + 3)B - script O sign(B^{-1}) except in intervals of width script O sign(B^{-1}) about each lower Landau level E_{m}(B) ≡ (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 energies E_{n}(B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.

Original language | English |
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Pages (from-to) | 355-369 |

Number of pages | 15 |

Journal | Letters in Mathematical Physics |

Volume | 40 |

Issue number | 4 |

DOIs | |

State | Published - Jun 1997 |

### Bibliographical note

Funding Information:J.M.C. is supported in part by CNRS and P.D.H. is supported in part by NSF grants INT 90-15895 and DMS 93-07438.

## Keywords

- Landau Hamiltonians
- Localization
- Random operators

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics