99-204 Vojkan Jaksic and Yoram Last

Spectral Structure of Anderson Type Hamiltonians (424K, postscript) Jun 1, 99

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Abstract. We study self adjoint operators of the form $H_{\omega} = H_0 + \sum \lambda_{\omega}(n) \scp{\delta_n}{\cdot} \delta_n$, where the $\delta_n$'s are a family of orthonormal vectors and the$\lambda_{\omega}(n)$'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 $\delta_n$ and $\delta_m$ are not completely orthogonal, then the restrictions of $H_{\omega}$ 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.

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