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

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