Simplicity of singular spectrum in anderson-type hamiltonians

Vojkan Jakšić*, Yoram Last

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

Abstract

We study self-adjoint operators of the form Hω = H 0 + ∑ 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.

Original languageAmerican English
Pages (from-to)185-204
Number of pages20
JournalDuke Mathematical Journal
Volume133
Issue number1
DOIs
StatePublished - 15 May 2006

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