A Contribution to the Sturmian Theory for Second-Order Non-Symmetric Systems

This thesis is divided into three chapters, each corresponding to one of the three papers Pertotti [11], [12] and Pertotti-Geaman [13]. The starting point of our research is a deep study of the "elementary" algebraic properties of X.4 and Y-t and their interelationships. Perhaps it is worth while remarking that the dimension of the vector space of n x n matrices is n 2 ,so that the dimension of the space of solutions to (2) is 2n2 • This means that, in the matrix case, XA. and Y-t do not constitute a basis for the space of solution, contrarily to the scalar case. These properties of X.-1. and Y-t are used in connection with ordered Banach space techniques (not only the Krein-Rutman theory!) in the proofs of the main results. These are the following: the existence of solutions whose components have a prescribed sign; - in ch.2, we characterize the existence of symmetric matrix solutions to (2). This is of interest in the oscillation theory and in the construction of the Bohl transformation, cf. Goff-St.Mary [5]; - in ch.3, we reduce the question of the existence of conjugate and focal points to the analysis of a scalar equation, in analogy with the Liouville theorem about the vVronskian of a matrix solution.