Fredholm property and essential spectrum of $3-D$ Dirac operators with regular and singular potentials

Abstract

We consider the $3-D$ Dirac operator $\mathfrak{D}_{\boldsymbol{A},\Phi ,Q_{\sin }}$ with variable regular magnetic and electrostatic potentials $ \boldsymbol{A}$,$\Phi $ and with singular potentials $Q_{\sin }$ with support on a smooth unbounded surface $\Sigma \subset \mathbb{R}^{3}$ which divides $\mathbb{R}^{3}$ on two open domains $\Omega_{\pm }$. We associate with the formal Dirac operator $\mathfrak{D}_{\boldsymbol{A},\Phi ,Q_{\sin }} $ an unbounded operator $\mathcal{D}_{\boldsymbol{A},\Phi ,Q_{\sin }}$ in $ L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$ generated by the regular part of $ \mathfrak{D}_{\boldsymbol{A},\Phi ,Q_{\sin }}$ with domain in $H^{1}(\Omega_{+},\mathbb{C}^{4})\oplus H^{1}(\Omega_{-},\mathbb{C}^{4})$ consisting of functions satisfying transmission conditions on $\Sigma .$ We consider the self-adjointness of operator $\mathcal{D}_{\boldsymbol{A},\Phi ,Q_{\sin }}$ for unbounded $C^{2}-$uniformly regular surfaces $\Sigma ,$ and the essential spectrum of $\mathcal{D}_{\boldsymbol{A},\Phi ,Q_{\sin }}$ if $ \Sigma $ is a $C^{2}$-surfaces with conic exits to infinity. As application we consider the electrostatic and Lorentz scalar $\delta_{\Sigma }-$shell interactions on unbounded surfaces $\Sigma .$