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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.

39A12Discrete version of topics in analysis
37J05Relations of dynamical systems with symplectic geometry and topology