Approximation of Dirac operators with $\boldsymbol{\delta}$-shell potentials in the norm resolvent sense, II. Quantitative results

This paper is devoted to the approximation of two and three-dimensional Dirac operators $H_{\widetilde{V} \delta_\Sigma}$ with combinations of electrostatic and Lorentz scalar $\delta$-shell interactions in the norm resolvent sense. Relying on results from \cite{BHS23} an explicit smallness condition on the coupling parameters is derived so that $H_{\widetilde{V} \delta_\Sigma}$ is the limit of Dirac operators with scaled electrostatic and Lorentz scalar potentials. Via counterexamples it is shown that this condition is sharp. The approximation of $H_{\widetilde{V} \delta_\Sigma}$ for larger coupling constants is achieved by adding an additional scaled magnetic term.