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

TL;DR

This paper provides a quantitative analysis of how two- and three-dimensional Dirac operators with delta-shell interactions can be approximated by scaled potentials, establishing explicit conditions and sharpness of the approximation.

Contribution

It introduces explicit smallness conditions for the coupling parameters ensuring norm resolvent convergence and extends the approximation to larger couplings with magnetic terms.

Findings

Derived explicit smallness conditions for coupling parameters.

Proved the sharpness of these conditions via counterexamples.

Extended approximation method to larger couplings with magnetic terms.

Abstract

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.