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.
Highlights
In this paper, we consider the symmetric three-term recurrence equationSk 1xk 2 − Tk 1xk 1 STk xk 0, k ∈ 0, N − 1 Z, T where xk ∈ Rn for k ∈ 0, N 1 Z, the real n × n matrices Sk and Tk are defined on0, N Z with Tk being symmetric and Sk being invertible
In 6, the authors proposed to study the Jacobi equations J as discrete symplectic systems S in a direct way which bypasses the Hamiltonian system H. This new approach requires that only the matrices Sk be invertible while the matrices Rk are allowed to be singular, which yields more general results for J obtained, for example, through the theory of symplectic systems S. We continue in this direction and we show that the three-term recurrence equations T naturally possess a symplectic structure Theorem 3.1 and Corollary 3.2
The matrices Pk, Qk, Rk, and the vectors xk in J have the following properties: Pk, Qk, Rk ∈ Rn×n are defined on 0, N Z, Pk and Rk are symmetric, and the matrix Sk : Rk QkT is invertible; xk ∈ Rn for k in 0, N 1 Z
Summary
Sk 1xk 2 − Tk 1xk 1 STk xk 0, k ∈ 0, N − 1 Z, T where xk ∈ Rn for k ∈ 0, N 1 Z, the real n × n matrices Sk and Tk are defined on. In 6 , the authors proposed to study the Jacobi equations J as discrete symplectic systems S in a direct way which bypasses the Hamiltonian system H This new approach requires that only the matrices Sk be invertible while the matrices Rk are allowed to be singular, which yields more general results for J obtained, for example, through the theory of symplectic systems S. The general theory of discrete symplectic systems recently developed, for example, in 7–18 can be applied to obtain, in particular, the Riccati equations and inequalities, and the oscillation and Sturmian theorems including multiplicities of focal points for the symmetric three-term recurrence equations T.
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