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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513774533

Digital Object Identifier
doi:10.2140/apde.2018.11.705

Mathematical Reviews number (MathSciNet)
MR3738260

Zentralblatt MATH identifier
06820937

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.apde/1513774533


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