On Dirac operators with electrostatic \(\delta \)-shell interactions of critical strength (Q2179894)

Summary: 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.