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 the free Dirac operator $\mathcal{H}$ coupled with the singular potential $V_\kappa=(\epsilon I_4 +\mu\beta+\eta(\alpha\cdot N))\delta_{\partial\Omega}$. The open set $\Omega$ can be either a $\mathcal{C}^2$-bounded domain or a locally deformed half-space. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of $\mathcal{H}+V_\kappa$ for the so-called critical combinations of coupling constants, when $\Omega$ is a locally deformed half-space. Finally, we introduce a new model of Dirac operators with $\delta$-interactions and deals with its spectral properties. More precisely, we study the coupling $\mathcal{H}_{\upsilon}=\mathcal{H}+i\upsilon\beta(\alpha\cdot N)\delta_{\partial\Omega}$. In particular, we show that $\mathcal{H}_{\pm2}$ is essentially self-adjoint and generates confinement.