Oscillation and spectral theory for symplectic difference systems with separated boundary conditions

Abstract

We consider symplectic difference systems involving a spectral parameter together with general separated boundary conditions. We establish the so-called oscillation theorem which relates the number of finite eigenvalues less than or equal to a given number to the number of focal points of a certain conjoined basis of the symplectic system. Then we prove Rayleigh's principle for the variational description of finite eigenvalues and we describe the space of admissible sequences by means of the (orthonormal) system of finite eigenvectors. The principle role in our treatment is played by the construction where the original system with general separated boundary conditions is extended to a system on a larger interval with Dirichlet boundary conditions.