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On the difference equations with periodic coefficients
Mathematical Physics · Submitted 13 Jun 2002
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the monodromy matrices corresponding to the minimal entire solutions are trigonometric polynomials of the same order as $M$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we study entire solutions of the difference equation $\psi(z+h)=M(z)\psi(z)$, $z\in{\mathbb C}$, $\psi(z)\in {\mathbb C}^2$. | contribution 0.86 | 0.89 | 0.04 | asserted 0.86 |
| s1 | In this equation, $h$ is a fixed positive parameter and $M: {\mathbb C}\to SL(2,{\mathbb C})$ is a given matrix function. | background 0.96 | 0.12 | 0.02 | asserted 1.00 |
| s2 | We assume that $M(z)$ is a $2\pi$-periodic trigonometric polynomial. | background 0.65 | 0.26 | 0.03 | asserted 1.00 |
| s3 | 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$z\to-\infty$. | contribution 0.90 | 0.96 | 0.24 | asserted 1.00 |
| s4 | We show that the monodromy matrices corresponding to the minimal entire solutions are trigonometric polynomials of the same order as $M$. | result 1.00 | 0.94 | 0.40 | asserted 1.00 |
| s5 | This property relates the spectral analysis of difference Schr\"odinger equations with trigonometric polynomial coefficients to an analysis of finite dimensional dynamical systems. | interpretation 1.00 | 0.57 | 0.43 | asserted 0.98 |
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Original 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.
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