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.