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

This paper is devoted to the approximation of two and three-dimensional Dirac operators $H_{\tilde{V}{\delta }_{\mathrm{\Sigma }}}$ 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 $H_{\tilde{V}{\delta }_{\mathrm{\Sigma }}}$ 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 $H_{\tilde{V}{\delta }_{\mathrm{\Sigma }}}$ for larger coupling constants is achieved by adding an additional scaled magnetic term.