Smooth approximation of singular perturbations

We consider a certain subclass of self-adjoint extensions of the symmetric operator −Δ|C∞0 R− {S} , where S ⊂ R, that correspond formally to perturbations of the Laplacian by potentials involving the δ-potential.We show that these extensions can be approximated in the strong resolvent sense by smooth perturbations of the Laplacian when S is both a finite and infinite subset of R. Also, we show that the operator in the finitely-many potential case approaches the operator in the infinitely-many potential case as the number of potentials approaches infinity. These results extend and unify what has previously been known about smooth approximations of point interactions in one dimension.  2003 Elsevier Inc. All rights reserved