#### Vol. 11, No. 3, 2018

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

### Albert Mas and Fabio Pizzichillo

Vol. 11 (2018), No. 3, 705–744
##### 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.

##### Keywords
Dirac operator, Klein's paradox, $\delta$-shell interaction, singular integral operator, approximation by scaled regular potentials, strong resolvent convergence
##### Mathematical Subject Classification 2010
Primary: 81Q10
Secondary: 35Q40, 42B20, 42B25
##### Milestones
Revised: 14 September 2017
Accepted: 16 October 2017
Published: 22 November 2017
##### Authors
 Albert Mas Departament de Matemàtiques i Informàtica Universitat de Barcelona Barcelona Spain Fabio Pizzichillo Basque Center for Applied Mathematics (BCAM) Bilbao Spain