Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 3 (2018), 705-744.

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

Albert Mas and Fabio Pizzichillo

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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.

Article information

Anal. PDE, Volume 11, Number 3 (2018), 705-744.

Received: 23 January 2017
Revised: 14 September 2017
Accepted: 16 October 2017
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
Secondary: 35Q40: PDEs in connection with quantum mechanics 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory

Dirac operator Klein's paradox $\delta$-shell interaction singular integral operator approximation by scaled regular potentials strong resolvent convergence


Mas, Albert; Pizzichillo, Fabio. Klein's paradox and the relativistic $\delta$-shell interaction in $\mathbb{R}^3$. Anal. PDE 11 (2018), no. 3, 705--744. doi:10.2140/apde.2018.11.705.

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