*(English)*Zbl 0942.39009

The authors present a survey on recent results connected to the discrete linear vector Hamiltonian system

where $B\left(t\right)$ and $C\left(t\right)$ are $n\times n$ Hermitian matrix functions on the discrete interval $[a,b]:=\{a,a+1,\cdots ,b\},a,b\in \mathbb{Z}$, and it is assumed that $I-A\left(t\right)$ is nonsingular on $[a,b]$.

The paper is organized as follows. Section 2 introduces linear Hamiltonian difference systems and explains how they are special cases of symplectic difference systems. Section 3 gives an introduction to symplectic systems and introduces important notions such as conjoined bases, generalized zeros, focal points, and disconjugacy.

In Section 4 are introduced the concept and provided the motivation of trigonometric systems. An explicit transformation is also offered which transforms any symplectic system into a trigonometric system and preserves oscillatory behavior. Section 6 presents an overview of the discrete calculus of variations. In Section 7 an introduction is given to the theory of Stefan Higler in order to unify discrete and continuous analysis.

##### MSC:

39A12 | Discrete version of topics in analysis |

37J05 | Relations of dynamical systems with symplectic geometry and topology |