The rigged hilbert spaces approach in singular perturbation theory

Abstract

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 0 and 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 à as the self-adjoint operator uniquely associated with a new rigged Hilbert space 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.