On the difference equations with periodic coefficients

Abstract

In this paper, we study entire solutions of the difference equation $\psi(z+h)=M(z)\psi(z)$, $z\in{\mathbb C}$, $\psi(z)\in {\mathbb C}^2$. In this equation, $h$ is a fixed positive parameter and $M: {\mathbb C}\to SL(2,{\mathbb C})$ is a given matrix function. We assume that $M(z)$ is a $2\pi$-periodic trigonometric polynomial. We construct the minimal entire solutions, i.e. entire solutions with the minimal possible growth simultaneously as for im$z\to+\infty$ so for im$z\to-\infty$. We show that the monodromy matrices corresponding to the minimal entire solutions are trigonometric polynomials of the same order as $M$. This property relates the spectral analysis of difference Schr\"odinger equations with trigonometric polynomial coefficients to an analysis of finite dimensional dynamical systems.