Spectral structure of Anderson type Hamiltonians

Vojkan Jakšić*, Yoram Last

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

44 Scopus citations


We study self adjoint operators of the form Hω = H0 + Σλω(n) 〈δn, ·〉δn, where the δn's are a family of orthonormal vectors and the λω(n)'s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n, m), if the cyclic subspaces corresponding to the vectors δn and δm are not completely orthogonal, then the restrictions of Hω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that "well behaved" absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases.

Original languageAmerican English
Pages (from-to)561-577
Number of pages17
JournalInventiones Mathematicae
Issue number3
StatePublished - 2000


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