Dirac operators, shell interactions and discontinuous gauge functions across the boundary

Abstract

Given a bounded smooth domain $\Omega\subset\mathbb{R}^3$, we explore the relation between couplings of the free Dirac operator $-i\alpha\cdot\nabla+m\beta$ with pure electrostatic shell potentials $\lambda\delta_{\partial\Omega}$ ($\lambda\in\mathbb{R}$) and some perturbations of those potentials given by the normal vector field $N$ on the shell $\partial\Omega$, namely $\{\lambda_e+\lambda_n(\alpha\cdot N)\}\delta_{\partial\Omega}$ ($\lambda_e$, $\lambda_n\in\mathbb{R}$). Under the appropiate change of parameters, the couplings with perturbed and unperturbed electrostatic shell potentials yield unitary equivalent self-adjoint operators. The proof relies on the construction of an explicit family of unitary operators that is well adapted to the study of shell interactions, and fits within the framework of gauge theory. A generalization of such unitary operators also allow us to deal with the self-adjointness of couplings of $-i\alpha\cdot\nabla+m\beta$ with some shell potentials of magnetic type, namely $\lambda(\alpha\cdot N)\delta_{\partial\Omega}$ with $\lambda\in C^1(\partial\Omega)$.