The complex WKB method for difference equations and Airy functions

TL;DR

This paper develops a complex WKB method to analyze solutions of a difference Schrödinger equation near turning points, providing uniform asymptotic expansions as the parameter h approaches zero.

Contribution

It introduces a novel complex WKB approach for difference equations, extending classical methods to discrete analogs and deriving uniform asymptotics near critical points.

Findings

Derived uniform asymptotic expansions near turning points.

Extended classical WKB methods to difference equations.

Analyzed solutions' behavior as h approaches zero.

Abstract

We consider the difference Schr{\"o}dinger equation $\psi$(z + h) + $\psi$(z -- h) + v(z)$\psi$(z) = 0 where z is a complex variable, h > 0 is a parameter, and v is an analytic function. As h $\rightarrow$ 0 analytic solutions to this equation have a standard quasiclassical behavior near the points where v(z) = $\pm$2. We study analytic solutions near the points z 0 satisfying v(z 0) = $\pm$2 and v (z 0) = 0. For the finite difference equation, these points are the natural analogues of the simple turning points defined for the differential equation --$\psi$ (z) + v(z)$\psi$(z) = 0. In an h-independent neighborhood of such a point, we derive uniform asymptotic expansions for analytic solutions to the difference equation.