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 $\eta \neq \pm 2$ 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.