Absolutely continuous spectrum of discrete Schrödinger operators with slowly oscillating potentials

We show that when a potential bn of a discrete Schrödinger operator, defined on l2(Z{double-struck}+), 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.

Duke Scholars

Author Alexander A. Kiselev Mathematics