Inventiones mathematicae

, Volume 141, Issue 3, pp 561–577

Spectral structure of Anderson type Hamiltonians

  • Vojkan Jakšić
  • Yoram Last

DOI: 10.1007/s002220000076

Cite this article as:
Jakšić, V. & Last, Y. Invent. math. (2000) 141: 561. doi:10.1007/s002220000076

Abstract.

We study self adjoint operators of the form¶Hω = H0 + ∑λω(n) <δn,·>δn,¶where the δn’s are a family of orthonormal vectors and the λω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δn and δm are not completely orthogonal, then the restrictions of Hω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Vojkan Jakšić
    • 1
  • Yoram Last
    • 2
  1. 1.Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, CanadaCA
  2. 2.Institute of Mathematics, The Hebrew University, 91904 Jerusalem, IsraelIL
  3. 3.Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USAUS