On monodromy matrices for a difference Schrödinger equation on the real line with a small periodic potential

In this paper one considers a one-dimensional difference Schrödinger equation $\psi(z+h) + \psi(z-h) + \lambda v(z) \psi(z) = E \psi(z) $ with a periodic potential $v$. In the case when the potential is real analytic, as well as in the case when, in a neighborhood of $\mathbb{R}$, the potential has a finite number of simple poles per period, for small values of the coupling constant $\lambda$, we describe the asymptotics of a monodromy matrix.