Non-local relativistic $\delta$-shell interactions

Abstract

In this paper, new self-adjoint realizations of the Dirac operator in dimension two and three are introduced. It is shown that they may be associated with the formal expression ${\mathcal{D}}_0+|F{\delta }_{\Sigma }⟩⟨G{\delta }_{\Sigma }|$, where ${\mathcal{D}}_0$ is the free Dirac operator, $F$ and $G$ are matrix valued coefficients, and ${\delta }_{\Sigma }$ stands for the single layer distribution supported on a hypersurface $\Sigma$, and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.