On the single layer boundary integral operator for the Dirac equation

TL;DR

This paper analyzes the mathematical properties of a singular boundary integral operator related to the Dirac equation, providing insights useful for understanding complex quantum interactions with singular potentials.

Contribution

It offers a detailed analysis of the single layer boundary integral operator for the Dirac equation, including its mapping properties and decomposition, applicable to singular interactions.

Findings

Decomposition of the operator into positive and negative parts

Mapping properties for smooth boundaries

Application to singular electrostatic and magnetic interactions

Abstract

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.