Smooth approximation of singular perturbations of the Laplacian

We consider a certain subclass of self-adjoint extensions of the symmetric operator −Δ|$C\infty 0$(R − {lcub}S{rcub}), where S ⊂ R, that correspond 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. We then prove the smooth approximation result in the relativistic case for the entire subclass with finitely-many potentials. These results extend and unify what has previously been known about smooth approximations of point interactions in one dimension.