Berry phase for difference equations

We study solutions to the difference equation Ψ(z + h) = M(z)Ψ(z), z ϵ C, where h > 0 is a small constant parameter and M : C → SL(2, C) is a given analytic matrix valued function. We describe the behavior of its analytic solutions as h → 0. The asymptotic formulas contain an analog of the Berry phase, well-known in the theory of differential equations.