Formula omitted]-self-adjoint extensions of a class of Hamiltonian differential systems

Abstract

This paper is concerned with J-self-adjoint extensions for a class of Hamiltonian differential systems. The domains of the corresponding minimal and maximal operators are described, a classification of limit types of the systems at each of endpoints is given, a property of the regularity field of the corresponding minimal operator is derived, and an expression of the J-defect index of the minimal operator by those of the left and right minimal operators is presented. Based on them, a characterization of all J-self-adjoint extensions for the Hamiltonian system is obtained in terms of square integrable solutions of a certain equation which is generated by this Hamiltonian system and its formal adjoint. As a consequence, characterizations of all J-self-adjoint extensions are given for the systems in some special cases. In particular, all J-self-adjoint extensions are characterized in terms of square integrable solutions of the system itself in certain special cases.