On the difference equations with periodic coefficients
| Autor: | Alexander Fedotov, Vladimir Buslaev |
|---|---|
| Rok vydání: | 2001 |
| Předmět: |
Physics
Pure mathematics Dynamical systems theory Mathematics - Complex Variables Differential equation 39B32 General Mathematics 39A10 FOS: Physical sciences General Physics and Astronomy Order (ring theory) Mathematical Physics (math-ph) Trigonometric polynomial Monodromy Matrix function FOS: Mathematics Spectral analysis Complex Variables (math.CV) Trigonometry Mathematical Physics |
| Zdroj: | Scopus-Elsevier |
| ISSN: | 1095-0753 1095-0761 |
| Popis: | 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. Comment: 45 pages |
| Databáze: | OpenAIRE |
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