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

TL;DR

This paper studies the behavior of the Dirac operator in three-dimensional space with a scaled potential, showing it converges to a $ abla$-shell interaction, revealing non-linear coupling effects related to Klein's Paradox.

Contribution

It demonstrates the strong resolvent convergence of the Dirac operator with scaled potential to a $ abla$-shell Hamiltonian, highlighting non-linear coupling dependence.

Findings

Convergence of Dirac operator to $ abla$-shell Hamiltonian

Non-linear dependence of coupling constant on potential

Klein's Paradox influences the interaction model

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.