Non-local relativistic $$\delta $$-shell interactions

Abstract

AbstractIn 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 $$\mathcal {D}_0+|F\delta _\Sigma \rangle \langle G\delta _\Sigma |$$ D 0 + | F δ Σ ⟩ ⟨ G δ Σ | , where $$\mathcal {D}_0$$ D 0 is the free Dirac operator, F and G are matrix valued coefficients, and $$\delta _\Sigma $$ δ Σ stands for the single layer distribution supported on a hypersurface $$\Sigma $$ Σ , and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.