Non-local relativistic $δ$-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 $\mathcal{D}0+|F\delta\Sigma\rangle\langle G\delta_\Sigma|$, 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.