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

Abstract

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