Complex WKB method for a system of two linear difference equations

Abstract

Analytic solutions of the difference equation Ψ ( z + h ) = M ( z ) Ψ ( z ) \Psi (z+h)=M(z)\Psi (z) are explored. Here z z is a complex variable, h > 0 h>0 is a parameter, and M M is a given S L ( 2 , C ) SL(2,\mathbb {C}) -valued function. It is assumed that M M either is analytic in a bounded domain or is a trigonometric polynomial. A simple method to derive the asymptotics of solutions as h → 0 h\to 0 is described.