## On the difference equations with periodic coefficients.(English)Zbl 1012.39008

Summary: 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 $$M:\mathbb{C}\mapsto SL(2,\mathbb{C})$$ is a given matrix function. We assume that $$M(z)$$ is a $$2\pi$$-periodic trigonometric polynomial. The main aim is to construct the minimal entire solutions, i.e. the solutions with the minimal possible growth simultaneously as for $$z\to-i \infty$$ so for $$z\to+i \infty$$.
We show that the monodromy matrices corresponding to the bases made of the minimal solutions are trigonometric polynomials of the same order as the matrix $$M$$. This property relates the spectral analysis of the one-dimensional difference Schrödinger equations with the potentials being trigonometric polynomials to an analysis of a finite dimensional dynamical system.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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