Complex WKB method for difference equations in bounded domains

We consider the difference Schrödinger equation $\psi(z+h)+\psi(z-h)+v(z)\psi(z)=E\psi(z)$, $z\in\mathbb C$, where $h>0$ and $E\in\mathbb C$ are parameters, and $v$ is a function analytic in a bounded domain $D\subset\mathbb C$. We develop an asymptotic method to study its solutons in the domain $D$ for small positive $h$.