RANK-ONE PERTURBATIONS OF NOT SEMIBOUNDED OPERATORS

Rank one perturbations of selfadjoint operators which are not necessar ily semibounded are studied in the present paper. It is proven that su ch perturbations are uniquely defined, if they are bounded in the sens e of forms. We also show that form unbounded rank one perturbations ca n be uniquely defined if the original operator and the perturbation ar e homogeneous with respect to a certain one parameter semigroup. The p erturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense withou t renormalization of the coupling constant only if the original operat or is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimensio n.