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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 $M:\mathbb{C}\mapstoSL(2,\mathbb{C})$ is a given analytic function. We describe the asymptotics ofits analytic solutions as $h\to 0$. The asymptotic formulas contain an analogof the geometric (Berry) phase well-known in the quasiclassical analysis ofdifferential equations.

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