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Discrete linear Hamiltonian systems: A survey. (English) Zbl 0942.39009
The authors present a survey on recent results connected to the discrete linear vector Hamiltonian system $$\Delta x(t)= A(t)x(t)+ B( t)u(t),\quad\Delta u(t)=C(t)x(t)-A^*(t)u(t),$$ where $B(t)$ and $C(t)$ are $n \times n$ Hermitian matrix functions on the discrete interval $[a,b]:= \{a,a+1, \dots,b\},a,b \in\bbfZ$, and it is assumed that $I-A(t)$ 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