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

Let _+^3 be a fixed bounded domain with boundary = +. We consider U^ a tubular neighborhood of the surface with a thickness parameter >0, and we define the perturbed Dirac operator D^₌=Dₘ +M 1ₔ^, with Dₘ the free Dirac operator, M>0, and 1ₔ ^ the characteristic function of U^. Then, in the norm resolvent sense, the Dirac operator D^M converges to the Dirac operator coupled with Lorentz scalar -shell interactions as = M^-1 tends to 0, with a convergence rate of O (M^-1).