Discrete Oscillation

Abstract

Dedicated to Allan Peterson on the Occasion of His 60th Birthday. In this paper we consider symplectic difference systems. The central aspect is the introduction of the concept of multiplicities of focal points (or of generalized zeros) of sequences solving these difference systems. This new notion makes it possible to count the number of focal points (of course, including multiplicities) in some interval. The main result concerns corresponding discrete eigenvalue problems. For any given eigenvalue parameter, we show that the number of eigenvalues, which are less than or equal to this parameter, equals the number of focal points of the principal solution of the symplectic difference system in the considered interval. This discrete oscillation theory includes Hamiltonian difference systems, in particular self-adjoint Sturm-Liouville difference equations of higher order. The results are in some sense discrete versions of well-known oscillation results for Hamiltonian differential systems.