Trigonometric transformations of symplectic difference systems (Q1566832)

The authors deal with the symplectic difference system \[ z_{k+1} ={\mathcal S}_kz_k, \tag{S} \] where \(z_k\in\mathbb{R}^{2n}\) and for each \(k\in\mathbb{Z}\), \({\mathcal S}_k\in \mathbb{R}^{2n\times 2n}\) satisfies the relation \({\mathcal S}_k^T{\mathcal I} {\mathcal S}_k={\mathcal I}\), where \({\mathcal I}=\left( \begin{smallmatrix} 0 & I\\ -I & 0 \end{smallmatrix} \right)\), \(I\) being the identity matrix. System (S) is called a trigonometric symplectic difference system if in addition to the above hypotheses \({\mathcal I}^T{\mathcal S}_k {\mathcal I}={\mathcal S}_k\) for \(k\in\mathbb{Z}\). The main result of the paper shows that any symplectic difference system (S) can be transformed into a trigonometric symplectic difference system, using a transformation which preserves the oscillatory properties of the original equation. This result can be regarded as a discrete version of an earlier result of the second author concerning Hamiltonian differential systems. For certain trigonometric difference systems the authors establish a necessary and sufficient condition to be nonoscillatory, and this result is applied to the oscillation theory of symplectic difference and Hamiltonian systems.