The Complex WKB Method for Difference Equations and Airy Functions

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.