01-216 Vladimir Buslaev, Alexander Fedotov

On the difference equations with periodic coefficients (149K, LaTeX 2e with 2 PS Figures) Jun 13, 01

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Abstract. In the paper, we study entire solutions of the diffrence equation % \begin{equation}\psi\,(z+h)=M\,(z)\,\psi\,(z),\quad z\in\C,\quad \psi\,(z)\in\C^2. \label{1.1}\end{equation} % In this equation, $h$ is a fixed positive parameter, and $M$ is a given matrix. We assume that the matrix $M$ satisfies the following two conditions. First, $M\,(z)\in SL\,(2,\,\C)$, $z\in\C$, secondly, the matrix $M$ is a $2\pi$-periodic trigonometric polynomial. The main aim is to construct the minimal entire solutions, e.i. 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 have the most simple analytic structure: they are trigonometric polynomials of the same order as the matrix $M$. This property is important for the spectral analysis of the one dimensional difference Schr\"odinger equations with the potentials being trigonometric polynomials. It relates their spectral analysis to an analysis of a finite dimensinal dynamical system.

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