Complex WKB method for a system of two linear difference equations

Abstract

Analytic solutions of the difference equationΨ(z+h)=M(z)Ψ(z)\Psi (z+h)=M(z)\Psi (z)are explored. Herezzis a complex variable,h>0h>0is a parameter, andMMis a givenSL(2,C)SL(2,\mathbb {C})-valued function. It is assumed thatMMeither is analytic in a bounded domain or is a trigonometric polynomial. A simple method to derive the asymptotics of solutions ash→0h\to 0is described.