R.Hilscher
A unified approach to continuous and discrete linear Hamiltonian systems
via the calculus on time scales
Abstract
We study a linear Hamiltonian system (H)
on an arbitrary time scale T, which allows (among others)
- to treat both continuous and discrete linear Hamiltonian systems
(as the special cases for T=R and T=Z) within one theory;
- to explain the discrepancies between these two theories while studying
systems of the form (H).
We derive the corresponding Wronskian identity, Riccati equation and
Picone identity on T. Transformations of the system (H) are studied as well.
As a main result we prove a sufficient condition for positive definiteness
of the quadratic functional associated with (H). Since we allow the matrix
B to be singular, the important Sturm-Liouville equations of higher order
may be studied as a special case of the system (H).