Non-local relativistic \(\delta \)-shell interactions

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 \(\mathcal {D}_0+|F\delta _\Sigma \rangle \langle G\delta _\Sigma |\) ${\mathcal{D}}_0+|F{\delta }_{\mathrm{\Sigma }}⟩⟨G{\delta }_{\mathrm{\Sigma }}|$ , where \(\mathcal {D}_0\) ${\mathcal{D}}_0$ is the free Dirac operator, F and G are matrix valued coefficients, and \(\delta _\Sigma \) ${\delta }_{\mathrm{\Sigma }}$ stands for the single layer distribution supported on a hypersurface \(\Sigma \) $\mathrm{\Sigma }$ , and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.