Hörmander's index and oscillation theory

Abstract

One of the obstacles that arises in the generalization of Sturm's oscillation theorem to the case of general linear Hamiltonian systems is the need to associate a sign with each crossing point, necessitating a signed count rather than a direct count of such points. It's known, however, that this difficulty does not arise for all system/boundary-condition combinations, and so it can be overcome in some cases by exchanging one boundary condition for another while also keeping track of any ancillary counts that accompany the exchange. The primary tool for making such an exchange is Hörmander's index, and in this analysis we develop a straightforward method for computing Hörmander's index, and employ our method to formulate oscillation-type theorems for linear Hamiltonian systems on both bounded and unbounded intervals.