Absolutely continuous spectrum of discrete Schrödinger operators with slowly oscillating potentials

We show that when a potential b n of a discrete Schrödinger operator, defined on l 2 (ℤ + ), slowly oscillates satisfying the conditions b n ∈ l ∞ and ∂ b n = b n +1 – b n ∈ l p , p < 2, then all solutions of the equation Ju = Eu are bounded near infinity at almost every E ∈ [–2 + lim sup n →∞ b n , 2 + lim sup n →∞ b n ] ∩ [–2 + lim inf n →∞ b n , 2 + lim inf n →∞ b n ]. We derive an asymptotic formula for generalized eigenfunctions in this case. As a corollary, the absolutely continuous spectrum of the corresponding Jacobi operator is essentially supported on the same interval of E (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)