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

Under certain hypotheses of smallness on the regular potential V, we prove that the Dirac operator in 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 C2 surface. Nevertheless, the coupling constant depends nonlinearly on the potential V; Klein’s paradox comes into play.

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
Received: 23 January 2017
Revised: 14 September 2017
Accepted: 16 October 2017
Published: 22 November 2017
Albert Mas
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Fabio Pizzichillo
Basque Center for Applied Mathematics (BCAM)