SPECTRAL PROPERTIES OF THE DIRAC OPERATOR COUPLED WITH δ-SHELL INTERACTIONS

Let $\O\subset\mathbb{R}^3$ 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 $V_\kappa=(\epsilon \mathit{I}_4 +\mu\beta + \eta(\alpha\cdot \mathit{N}))\delta_{\partial\Omega}$ , where $\kappa=(\epsilon,\mu,\eta)\in\mathbb{R}^3$. In the first instance, $\O$ can be either a $\mathcal{C}^2$-bounded domain or a locally deformed half-space. In both cases, the self-adjointness is proved and several spectral properties are given. In particular, we extend the result of \cite{BHOP} to the case of a locally deformed half-space, by giving a complete description of the essential spectrum of $ \mathcal{H}+V_\k $, for the so-called critical combinations of coupling constants. In the second part of the paper, the case of bounded rough domains is investigated. Namely, in the non-critical case and under the assumption that $\O$ has a $\mathrm{VMO}$ normal, we show that $ \mathcal{H}+V_\kappa $ is still self-adjoint and preserves almost all of its spectral properties. More generally, under certain assumptions about the sign or the size of the coupling constants, we are able to show the self-adjointness of the coupling $ \mathcal{H} + (\epsilon I_4 +\mu\beta )\delta_{\partial\Omega}$ , when $\Omega$ is bounded uniformly rectifiable. Moreover, if $\epsilon^2-\mu^2=-4$, we then show that $\partial\O$ is impenetrable. In particular, if $\Omega$ is Lipschitz, we then recover the same spectral properties as in the VMO case. In addition, we establish a characterization of regular Semmes-Kenig-Toro domains via the compactness of the anticommutator between $(\alpha\cdot \mathit{N})$ and the Cauchy operator associated to the free Dirac operator. Finally, we study the coupling $\mathcal{H}_{\upsilon}=\mathcal{H}+ i\u\beta(\alpha\cdot N)\delta_{\partial\Omega}$. In particular, if $\Omega$ is a bounded $\mathcal{C}^2$ domain, then we show that $\mathcal{H}_{ \pm2}$ is essentially self-adjoint and generates confinement