Vojkan Jaksic and Yoram Last
Spectral Structure of Anderson Type Hamiltonians
(424K, postscript)
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.