Rank One Perturbations, Approximations, and Selfadjoint Extensions

Rank-one perturbationsAα=A+α(ϕ, ·) ϕof a semibounded selfadjoint operatorAare studied with the help of distribution theory. It is shown that such perturbations can be defined for finite values ofαeven if the elementϕdoes not belong to H−1(A). Approximations of the rank one perturbations are constructed in the strong operator topology. It is proven that rank one H−2perturbations can be defined uniquely for the homogeneous operators. The results are applied to a Schrödinger operator with a delta interaction in dimension 3.