Klein’s paradox and the relativistic δ-shell interaction in ℝ3

Mas, Albert; Pizzichillo, Fabio

doi:10.2140/apde.2018.11.705

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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

DOI: 10.2140/apde.2018.11.705

Abstract

Under certain hypotheses of smallness on the regular potential $V$ , we prove that the Dirac operator in $ℝ^{3}$ , coupled with a suitable rescaling of $V$ , converges in the strong resolvent sense to the Hamiltonian coupled with a $δ$ -shell potential supported on $Σ$ , a bounded $C^{2}$ surface. Nevertheless, the coupling constant depends nonlinearly on the potential $V$ ; 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

Received: 23 January 2017

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

Vol. 11, No. 3, 2018