In this paper we study eigenvalue and oscillation properties of time scale symplectic systems with Dirichlet boundary conditions. The focus is on deriving the so-called oscillation theorems for these systems, which relate the number of finite eigenvalues of the system with the number of proper (or generalized) focal points of the principal solution of the system. This amounts to defining and developing the central notions of finite eigenvalues and proper focal points for the time scale environment. We establish the traditional geometric properties of finite eigenvalues and eigenfunctions enjoyed by self-adjoint linear systems. We assume no controllability or normality of the system.

Last change: December 9, 2010. (c) Roman Simon Hilscher