TY - JOUR

T1 - Spectral structure of Anderson type Hamiltonians

AU - Jakšić, Vojkan

AU - Last, Yoram

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034349435&partnerID=8YFLogxK

U2 - 10.1007/s002220000076

DO - 10.1007/s002220000076

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AN - SCOPUS:0034349435

SN - 0020-9910

VL - 141

SP - 561

EP - 577

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -