Generalized focal points and local Sturmian theory for linear Hamiltonian systems

In this paper we present a new approach for the study of the oscillation properties of linear differential equations, in particular of linear Hamiltonian systems. We introduce a new notion of a generalized left focal point as well as its multiplicity, which do not depend on the validity of the traditionally assumed Legendre condition. Based on this notion we are able to develop a local (or pointwise) version of the Sturmian separation theorem, which provides a lower bound and an upper bound for the multiplicity of a generalized left focal point for any conjoined basis of the system. We apply this knowledge in several directions, such as (ⅰ) in the explanation of the exact role of the Legendre condition in the Sturmian theory, (ⅱ) in the second order optimality conditions for variational problems, (ⅲ) in the analysis of isolated and non-isolated generalized left focal points, and (ⅳ) in the study of the so-called anti-Legendre condition. As a main tool we use the comparative index and its properties. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm–Liouville differential equations of arbitrary even order.