Klein's Paradox and the Relativistic $\delta$-shell Interaction in $\mathbb{R}^3$

Abstract

Under certain hypothesis of smallness of the regular potential $\mathbf{V}$, we prove that the Dirac operator in $\mathbb{R}^3$ coupled with a suitable re-scaling of $\mathbf{V}$ converges in the strong resolvent sense to the Hamiltonian coupled with a $\delta$-shell potential supported on $\Sigma$, a bounded $C^2$ surface. Nevertheless, the coupling constant depends non-linearly on the potential $\mathbf{V}$: the Klein's Paradox comes into play.