Finite-Rank Perturbations of the Dirac Operator

Abstract

In this paper we determine those relativistic point interactions for which renormalization of the coupling constant occurs when the corresponding potentials are approximated by local, short-range perturbations of the free Dirac operator in one dimension. We also find a general formula for the renormalization constant for the entire four-parameter family of relativistic point interactions. Our results, which unify and extend earlier work on relativistic point interactions, include perturbations of the Dirac operator by finitely many δ or δ′ potentials, as well as relativistic scalar and magnetic point interactions. In addition, we show that similar methods may be used to define more general perturbations of the Dirac operator, including time-dependent ones. These operators are shown to depend continuously on the perturbation in the sense of strong resolvent convergence.