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Symmetric Three-Term Recurrence Equations and Their Symplectic Structure

Abstract

We revive the study of the symmetric three-term recurrence equations. Our main result shows that these equations have a natural symplectic structure, that is, every symmetric three-term recurrence equation is a special discrete symplectic system. The assumptions on the coefficients in this paper are weaker and more natural than those in the current literature. In addition, our result implies that symmetric three-term recurrence equations are completely equivalent with Jacobi difference equations arising in the discrete calculus of variations. Presented applications of this study include the Riccati equation and inequality, detailed Sturmian separation and comparison theorems, and the eigenvalue theory for these three-term recurrence and Jacobi equations.

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Correspondence to Roman Šimon Hilscher.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Šimon Hilscher, R., Zeidan, V. Symmetric Three-Term Recurrence Equations and Their Symplectic Structure. Adv Differ Equ 2010, 626942 (2010). https://doi.org/10.1155/2010/626942

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  • DOI: https://doi.org/10.1155/2010/626942

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation