## Duke Mathematical Journal

### Simplicity of singular spectrum in Anderson-type Hamiltonians

#### 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 that provides in this setting a natural decomposition of the Hilbert space as a direct sum of mutually orthogonal closed subspaces, which are a.s. invariant under $H_\omega$, and that is helpful for the spectral analysis of such operators. We then use this decomposition to prove that the singular spectrum of $H_\omega$ is a.s. simple

#### Article information

Source
Duke Math. J., Volume 133, Number 1 (2006), 185-204.

Dates
First available in Project Euclid: 19 April 2006

https://projecteuclid.org/euclid.dmj/1145452059

Digital Object Identifier
doi:10.1215/S0012-7094-06-13316-1

Mathematical Reviews number (MathSciNet)
MR2219273

Zentralblatt MATH identifier
1107.47027

#### Citation

Jakšić, Vojkan; Last, Yoram. Simplicity of singular spectrum in Anderson-type Hamiltonians. Duke Math. J. 133 (2006), no. 1, 185--204. doi:10.1215/S0012-7094-06-13316-1. https://projecteuclid.org/euclid.dmj/1145452059

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