Spectral properties of the Dirac operator coupled with $$\delta $$-shell interactions

Let $$\Omega \subset {{\mathbb {R}}}^3$$ be an open set. We study the spectral properties of free Dirac operator $$ \mathcal {H} :=- i \alpha \cdot \nabla + m\beta coupled with singular potential $$V_\kappa =(\epsilon I_4 +\mu \beta \eta (\alpha N))\delta _{\partial \Omega }$$ , where $$\kappa ,\mu ,\eta )\in . The set can either a $$\mathcal {C}^2$$ -bounded domain or locally deformed half-space. In both cases, self-adjointness is proved and several are given. particular, we give complete description essential spectrum {H}+V_\kappa in case half-space, for so-called critical combinations coupling constants. Finally, introduce new model operators $$\delta -interactions deal its properties. More precisely, {H}_{\zeta ,\upsilon }=\mathcal {H}+ \left( -i\zeta _1\alpha _2\alpha _3+ i\upsilon N\right) \right) \delta $$\zeta \in {R}}}$$ show that {H}_{ 0,\pm 2}$$ essentially self-adjoint generates confinement.