Klein's paradox and the relativistic $\delta$-shell interaction in $\mathbb{R}^3$

Abstract

Under certain hypotheses of smallness on the regular potential $\mathbf{V}$, we prove that the Dirac operator in ${ℝ}^3$, coupled with a suitable rescaling 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 nonlinearly on the potential $\mathbf{V}$; Klein’s paradox comes into play.