Landau Hamiltonians with Unbounded Random Potentials

J. M. Barbaroux, J. M. Combes, P. D. Hislop

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9 Scopus citations


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 supλ∈ℝ3+∈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 Bn, such that for all B > Bn, 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 Em(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 En(B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.

Original languageEnglish
Pages (from-to)355-369
Number of pages15
JournalLetters in Mathematical Physics
Issue number4
StatePublished - 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.


  • Landau Hamiltonians
  • Localization
  • Random operators

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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