On the single layer boundary integral operator for the Dirac equation

This paper is devoted to the analysis of the single layer boundary integral operator ${\mathcal{C}}_z$ for the Dirac equation in the two- and three-dimensional situation. The map ${\mathcal{C}}_z$ is the strongly singular integral operator having the integral kernel of the resolvent of the free Dirac operator $A_0$ and $z$ belongs to the resolvent set of $A_0$. In the case of smooth boundaries fine mapping properties and a decomposition of ${\mathcal{C}}_z$ in a ‘positive’ and ‘negative’ part are analyzed. The obtained results can be applied in the treatment of Dirac operators with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions that are combined in a critical way.