A characterization of self-adjoint operators determined by the weak formulation of second-order singular differential expressions

In this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression φ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from φ by integration by parts once agrees with the inner product 〈φu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).