Complex WKB method for difference equations in unbounded domains

We study entire solutions to the difference Schrödinger equation ψ(z + h) + ψ(z - h) + v(z)ψ(z) = Eψ(z), z ϵ ℂ, where h > 0 and E ϵ ℂ are parameters, and v is a trigonometric polynomial. We describe asymptotic behavior of the solutions as h tends to zero.