Limit point, strong limit point and Dirichlet conditions for Hamiltonian differential systems

This paper deals with singular Hamiltonian differential systems. Three conditions on the asymptotic behavior or square integrability of their maximal domain functions at a singular end point are studied: the limit point condition, the strong limit point condition and the Dirichlet condition. The equivalence between the limit point and strong limit point conditions is established for a class of such systems, and for another class, the three conditions are shown to imply each other. As an application, two unified descriptions of the Friedrichs extension for some systems in the second class are obtained. A key feature of the descriptions is: they do not use the deficiency indices of the systems. Several illustrating examples are presented. In particular, two simple descriptions of the Friedrichs extension for a family of Schrödinger operators with singular potentials are achieved. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim