Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

We consider random self-adjoint Jacobi matrices of the form (Jωu)(n) = an(ω)u(n+ 1) +bn(ω)u(n)+ an−1(ω)u(n −1) on 2(N), where {an(ω) > 0} and {bn(ω) ∈ R} are sequences of random variables on a probability space (Ω, dP(ω)) such that there exists q ∈ N, such that for any l ∈ N, β2l(ω) = al(ω) − al+q(ω) and β2l+1(ω) = bl(ω)− bl+q(ω) are independent random variables of zero mean satisfying ∞ n=1 Ω β2 n(ω) dP(ω) <∞. Let Jp be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in Jω. We prove that for a.e. ω, the a.c. spectrum of the operator Jω equals to and fills the spectrum of Jp. If, moreover, ∞ n=1 Ω β4 n(ω) dP(ω) <∞,then for a.e. ω, the spectrum of Jω is purely absolutely continuous on the interior of the bands that make up the spectrum of Jp. © 2010 Published by Elsevier Inc