Duke Mathematical Journal

Simplicity of singular spectrum in Anderson-type Hamiltonians

Vojkan Jakšić and Yoram Last

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Abstract

We study self-adjoint operators of the form Hω=H0+ω(n)(δn|·)δn, where the δn's are a family of orthonormal vectors and the ω(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 Hω, and that is helpful for the spectral analysis of such operators. We then use this decomposition to prove that the singular spectrum of Hω 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

Permanent link to this document
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

Subjects
Primary: 47B80: Random operators [See also 47H40, 60H25]
Secondary: 47B25: Symmetric and selfadjoint operators (unbounded) 47A10: Spectrum, resolvent 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 60H25: Random operators and equations [See also 47B80] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

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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