On Dirac operators with electrostatic $\delta$-shell interactions of critical strength

Abstract

In this paper we prove that the Dirac operator $A_{\eta }$ with an electrostatic $\delta$-shell interaction of critical strength $\eta =\pm 2$ supported on a $C^2$-smooth compact surface $\Sigma$ is self-adjoint in $L^2({\mathbb{R}}^3;{\mathbb{C}}^4)$, we describe the domain explicitly in terms of traces and jump conditions in $H^{-1/2}(\Sigma ;{\mathbb{C}}^4)$, and we investigate the spectral properties of $A_{\eta }$. While the non-critical interaction strengths $\eta \ne \pm 2$ have received a lot of attention in the recent past, the critical case $\eta =\pm 2$ remained open. Our approach is based on abstract techniques in extension theory of symmetric operators, in particular, boundary triples and their Weyl functions.