On asymptotic summation of potentially oscillatory difference systems

A new technique for the asymptotic summation of linear systems of difference equations Y(t + 1) = (D(t) + R(t))Y (t) is derived. A fundamental solution Y(t) = Φ(t)(I + P(t)) is constructed in terms of a product of two matrix functions. The first function Φ(t) is a product of the diagonal part D(t). The second matrix I + P(t), is a perturbation of the identity matrix I. Conditions are given on the matrix D(t) + R(t) that allow us to represent I + P(t) as an absolutely convergent resolvent series without imposing stringent conditions on R(t). Our method could be applied to discretized version of singularly perturbed differential equations Y (t) = A(t)Y(t) that fit the setting of quantum mechanics. © 2006 Elsevier Inc. All rights reserved.