In this paper we prove singular comparison theorems on unbounded intervals for two nonoscillatory linear Hamiltonian systems satisfying the Sturmian majorant condition and the Legendre condition. At the same time we do not impose any controllability condition. The results are phrased in terms of the comparative index and the numbers of proper focal points of the (minimal) principal solutions of these systems at both endpoints of the considered interval. The main idea is based on an application of new transformation theorems for principal and antiprincipal solutions at infinity and on new limit properties of the comparative index involving these solutions. This work generalizes the recently obtained Sturmian separation theorems on unbounded intervals for one system by the authors (2019), as well as the Sturmian comparison theorems and transformation theorems on compact intervals by J. Elyseeva (2016 and 2018). We note that all the results are new even in the completely controllable case.

Last change: 12.02.2020. (c) Roman Simon Hilscher