The complex WKB method for difference equations and Airy functions
Résumé
We consider the difference Schrödinger equation ψ(z + h) + ψ(z − h) + v(z)ψ(z) = 0 where z is a complex variable, h > 0 is a parameter, and v is an analytic function. As h → 0 analytic solutions to this equation have a standard quasiclassical behavior near the points where v(z) = ±2. We study analytic solutions near the points z 0 satisfying v(z 0) = ±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 −ψ (z) + v(z)ψ(z) = 0. In an h-independent neighborhood of such a point, we derive uniform asymptotic expansions for analytic solutions to the difference equation.
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