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

Abstract

We consider the $3-D$ Dirac operator ${\mathfrak{D}}_{\mathbf{A},\Phi ,Q_{\sin }}$ with variable regular magnetic and electrostatic potentials $\mathbf{A}$,$\Phi$ 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 ${\mathfrak{D}}_{\mathbf{A},\Phi ,Q_{\sin }}$ an unbounded operator ${\mathcal{D}}_{\mathbf{A},\Phi ,Q_{\sin }}$ in $L^2({\mathbb{R}}^3,{\mathbb{C}}^4)$ generated by the regular part of ${\mathfrak{D}}_{\mathbf{A},\Phi ,Q_{\sin }}$ 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 ${\mathcal{D}}_{\mathbf{A},\Phi ,Q_{\sin }}$ for unbounded $C^2-$uniformly regular surfaces $\Sigma ,$ and the essential spectrum of ${\mathcal{D}}_{\mathbf{A},\Phi ,Q_{\sin }}$ 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 $\Sigma .$

27 pages