This paper deals with the oscillation behavior of linear Hamiltonian systems and related eigenvalue problems with general, linearly independent self-adjoint boundary conditions. The main new aspect of this paper is the fact that we do not require controllability, strong observability or strong normality of the system. In view of this generalization it is necessary to introduce a new notion of “proper” eigenvalues and their multiplicities of the related eigenvalue problem. We show that the “proper” eigenvalues of the related eigenvalue problems are always isolated. Furthermore, we introduce a new notion of the multiplicity of a “proper” focal point of so-called conjoined bases of the differential system. We derive oscillation theorems which give a formula for the number of all “proper” eigenvalues (including multiplicities) smaller or equal than a certain constant with respect to the number of all “proper” focal points (including multiplicities) of a certain conjoined basis of the Hamiltonian system. Due to this generalization we are able to treat more general Sturm–Liouville eigenvalue problems as in the existing literature.
Linear Hamiltonian system, eigenvalue problem, focal point, generalized zero, Sturm-Liouville equation
