The rigged Hilbert spaces approach in singular perturbation theory

We discuss a new approach in singular perturbation theory which is based on the method of rigged Hilbert spaces. Let A be a self-adjoint unbounded operator in a state space H-o and H- ] H-o ] H+ be the rigged Hilbert space associated with A in the sense that dom A = H+ in the graph-norm. We propose to define the perturbed operator (A) over tilde as the self-adjoint operator uniquely associated with a new rigged Hilbert space (H) over tilde (-) ] H-o ] (H) over tilde (+) constructed using a given perturbation of A. We show that the well-known form-sum and self-adjoint extensions methods are included in the above construction. Moreover, we show that the super singular perturbations may also be described in our framework.