- 04-248 Vojkan Jaksic and Yoram Last
- Simplicity of singular spectrum in Anderson type Hamiltonians
Aug 7, 04
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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.