04-248 Vojkan Jaksic and Yoram Last

Simplicity of singular spectrum in Anderson type Hamiltonians (217K, pdf) 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.

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