Resolvent formulas of general type and its application to point interactions

Kiselev and Simon ([13]) considered rank one singular perturbations of general type and formulate such perturbation in terms of a resolvent formula. In an attempt to generalize it beyond the rank one case, it is found that this expression in its generality describes resolvents of a rather wide class of closed operators (or all selfadjoint operators). Bounded operators from the domain of unperturbed operator with the graph norm to its dual space will serve as a parameter. As an application point interactions in one-dimension will be discussed systematically, recapturing selfadjoint boundary conditions associated to the problem.