Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints ✩

This paper is concerned with self-adjoint extensions for a linear Hamiltonian system with two singular endpoints. The domain of the closure of the corresponding minimal Hamiltonian operator H0 is described by properties of its elements at the endpoints of the discussed interval, decompositions of the domains of the corresponding left and right maximal Hamiltonian operators are provided, and expressions of the defect indices of H0 in terms of those of the left and right minimal operators are given. Based on them, characterizations of all the self-adjoint extensions for a Hamiltonian system are obtained in terms of square integrable solutions. As a consequence, the characterizations of all the self-adjoint extensions are given for systems in several special cases. © 2010 Elsevier Inc. All rights reserved