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

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