Vojkan Jaksic and Yoram Last
Simplicity of singular spectrum in Anderson type Hamiltonians
(217K, pdf)

ABSTRACT.  We study self adjoint operators of the form 
$H_\omega=H_0 +\sum \omega(n)(\delta_n|\,\cdot\,)\delta_n$, 
where the $\delta_n$'s are a family of orthonormal vectors 
and the $\omega(n)$'s are independent random variables 
with absolutely continuous probability distributions. 
We prove a general structural theorem which provides 
in this setting a natural decomposition of the Hilbert space 
as a direct sum of mutually orthogonal closed subspaces 
that are almost surely invariant under $H_\omega$ and which 
is helpful for the spectral analysis of such operators. 
We then use this decomposition to prove that the 
singular spectrum of $H_\omega$ is almost surely simple.