Absolutely continuous spectrum of discrete Schrödinger operators with slowly oscillating potentials

Abstract

AbstractWe show that when a potential bn of a discrete Schrödinger operator, defined on l2(ℤ+), slowly oscillates satisfying the conditions bn ∈ l∞ and ∂bn = bn +1 – bn ∈ lp, p < 2, then all solutions of the equation Ju = Eu are bounded near infinity at almost every E ∈ [–2 + lim supn →∞ bn, 2 + lim supn →∞ bn ] ∩ [–2 + lim infn →∞ bn, 2 + lim infn →∞ bn ]. 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)