On the approximation of the Dirac operator coupled with confining Lorentz scalar $\delta$-shell interactions

Let ${\mathrm{\Omega }}_{+}\subset {\mathbb{R}}^3$ be a fixed bounded domain with boundary $\mathrm{\Sigma }=\mathrm{\partial }{\mathrm{\Omega }}_{+}$. We consider ${\mathcal{U}}^{\epsilon }$ a tubular neighborhood of the surface $\mathrm{\Sigma }$ with a thickness parameter $\epsilon >0$, and we define the perturbed Dirac operator ${\mathfrak{D}}_M^{\epsilon }=D_m+M\beta {\mathbb{1}}_{{\mathcal{U}}^{\epsilon }},$ with $D_m$ the free Dirac operator, $M>0$, and ${\mathbb{1}}_{{\mathcal{U}}^{\epsilon }}$ the characteristic function of ${\mathcal{U}}^{\epsilon }$. Then, in the norm resolvent sense, the Dirac operator ${\mathfrak{D}}_M^{\epsilon }$ converges to the Dirac operator coupled with Lorentz scalar $\delta$-shell interactions as $\epsilon =M^{-1}$ tends to $0$, with a convergence rate of $\mathcal{O}(M^{-1})$.