Smooth Approximation of Finitely-Many Relativistic Point Interactions

Abstract. The purpose of this paper is to show that finitely-many relativistic point interac-tions may be approximated in the strong resolvent sense by sequences of operators with smooth potentials. We consider the entire family of relativistic point interactions, and determine those for which renormalization of the coupling constant occurs when the corresponding potentials are approximated by finitely-many local, short-range perturbations of the free Dirac operator. 1 The study of point interactions has a venerable history, beginning with the work of Kronig and Penney [12], and supported by a vast literature that illustrates their wide range of applications and fundamental importance for physics and mathematics (cf. [1] for a comprehensive treatment, and [5], [6], [7], [13], and [16] for results in the nonrelativistic case.) Relativistic point interactions and, more generally, finite-rank singular perturbations of nonsemibounded self-adjoint operators, have been the focus of considerable recent attention (cf. [4]). Results on approximation by sequences of operators with regular perturbations have been obtained by Albeverio and Kurasov [2], Albeverio, Koshmanenko, Kurasov, an