 01216 Vladimir Buslaev, Alexander Fedotov
 On the difference equations with periodic
coefficients
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Jun 13, 01

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Abstract. In the paper, we study entire solutions of the diffrence equation
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\begin{equation}\psi\,(z+h)=M\,(z)\,\psi\,(z),\quad z\in\C,\quad
\psi\,(z)\in\C^2. \label{1.1}\end{equation}
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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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