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.