Oscillation theorems for linear Hamiltonian systems with nonlinear dependence on the spectral parameter and the comparative index

Abstract

We consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. In our consideration we do not impose any controllability and strict normality assumptions and omit the Legendre condition for the Hamiltonian. We adapt the theory developed by A.A. Abramov for the Hamiltonian spectral problems based on piecewise constant transformations of their conjoined bases using the notion of the comparative index. We introduce the concept of oscillation numbers which generalize the notion of proper focal points and prove the oscillation theorem relating the number of finite eigenvalues in the given interval with the values of the oscillation numbers at the end points of this interval.