Difference equations in the complex plane: quasiclassical asymptotics and Berry phase

We study solutions to the difference equation $\mathrm{\Psi }(z+h)=M(z)\mathrm{\Psi }(z)$ where $z$ is a complex variable, $h>0$ is a parameter, and $M:$\mathbb{C}$\mapsto SL(2,$\mathbb{C}$)$ is a given analytic function. We describe the asymptotics of its analytic solutions as $h\to 0$. The asymptotic formulas contain an analog of the geometric (Berry) phase well-known in the quasiclassical analysis of differential equations.