On the approximation of the Dirac operator coupled with confining Lorentz scalar δ-shell interactions

Let Ω₊³ be a fixed bounded domain with boundary Σ = ∂Ω₊. We consider U^ε a tubular neighborhood of the surface Σ with a thickness parameter ε>0, and we define the perturbed Dirac operator DεM=Dₘ +Mβ 1_Uε, with Dₘ the free Dirac operator, $M>0$, and 1_U ε the characteristic function of Uε. Then, in the norm resolvent sense, the Dirac operator DεM converges to the Dirac operator coupled with Lorentz scalar δ-shell interactions as ε = M⁻¹ tends to $0$, with a convergence rate of O(M⁻¹).