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

Abstract

We study solutions to the difference equation $\Psi(z+h)=M(z)\Psi(z)$ 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.