, Volume 141, Issue 3, pp 561-577

Spectral structure of Anderson type Hamiltonians

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Abstract.

We study self adjoint operators of the form¶H ω = H 0 + ∑λω(n) <δ n ,·>δ n ,¶where the δ n ’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 δ n and δ m are not completely orthogonal, then the restrictions of H ω 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.

Oblatum 27-V-1999 & 6-I-2000¶Published online: 8 May 2000