Comparison theorems for weighted focal points of conjoined bases of Hamiltonian difference systems

Abstract

In this paper we prove comparison theorems for the number of weighted focal points of conjoined bases of Hamiltonian difference systems. The notion of a weighted focal point introduced by O. Došlý and J. Elyseeva (Appl.Math. Lett. (43) 2015, 114–119) plays an important role in spectral theory for discrete Hamiltonian eigenvalue problems with nonlinear dependence on the spectral parameter. We present new relations between the numbers of weighted focal points of conjoined bases of two Hamiltonian systems and derive corollaries to these relations generalizing comparison results for the classical number of focal points. The consideration is based on the comparative index theory for symplectic difference systems.