On the approximation of the δ-shell interaction for the 3-D Dirac operator

Mahdi Zreik

doi:10.56754/0719-0646.2603.489

Authors

Mahdi Zreik Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux 33405 Talence Cedex, France – Departamento de Matemáticas, Universidad del País Vasco, Barrio Sarriena s/n 48940 Leioa, Spain, and Basque Center for Applied Mathematics (BCAM), Spain. https://orcid.org/0000-0002-9891-0525

DOI:

https://doi.org/10.56754/0719-0646.2603.489

Keywords:

Dirac operators, self-adjoint operators, shell interactions, non critical and non-confining interaction strengths, approximations

Abstract

We consider the three-dimensional Dirac operator coupled with a combination of electrostatic and Lorentz scalar δ-shell interactions. We approximate this operator with general local interactions \(V\). Without any hypotheses of smallness on the potential \(V\), we investigate convergence in the strong resolvent sense to the Dirac Hamiltonian coupled with a δ-shell potential supported on \(S\), a bounded smooth surface. However, the coupling constant depends nonlinearly on the potential \(V\).

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References

N. Arrizabalaga, A. Mas, and L. Vega, “Shell interactions for Dirac operators,” J. Math. Pures Appl. (9), vol. 102, no. 4, pp. 617–639, 2014, doi: 10.1016/j.matpur.2013.12.00.

N. Arrizabalaga, A. Mas, and L. Vega, “Shell interactions for Dirac operators: on the point spectrum and the confinement,” SIAM J. Math. Anal., vol. 47, no. 2, pp. 1044–1069, 2015, doi: 10.1137/14097759X.

N. Arrizabalaga, A. Mas, and L. Vega, “An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators,” Comm. Math. Phys., vol. 344, no. 2, pp. 483–505, 2016, doi: 10.1007/s00220-015-2481-y.

J. Behrndt, M. Holzmann, and C. Stelzer, “Approximation of Dirac operators with δ-shell potentials in the norm resolvent sense,” 2023, arXiv:2308.13344.

J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik, “Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces,” Math. Nachr., vol. 290, no. 8-9, pp. 1215–1248, 2017, doi: 10.1002/mana.201500498.

J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik, “On Dirac operators in R3 with electrostatic and Lorentz scalar δ-shell interactions,” Quantum Stud. Math. Found., vol. 6, no. 3, pp. 295–314, 2019, doi: 10.1007/s40509-019-00186-6.

J. Behrndt, M. Holzmann, and M. Tušek, “Two-dimensional Dirac operators with general δ-shell interactions supported on a straight line,” J. Phys. A, vol. 56, no. 4, 2023, Art. ID 045201, doi: 10.1088/1751-8121/acafaf.

B. Cassano, V. Lotoreichik, A. Mas, and M. Tušek, “General δ-shell interactions for the two- dimensional Dirac operator: self-adjointness and approximation,” Rev. Mat. Iberoam., vol. 39, no. 4, pp. 1443–1492, 2023, doi: 10.4171/rmi/1354.

J. Dittrich, P. Exner, and P. Šeba, “Dirac operators with a spherically symmetric δ-shell interaction,” J. Math. Phys., vol. 30, no. 12, pp. 2875–2882, 1989, doi: 10.1063/1.528469.

L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, revised ed., ser. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.

R. J. Hughes, “Relativistic point interactions: approximation by smooth potentials,” Rep. Math. Phys., vol. 39, no. 3, pp. 425–432, 1997, doi: 10.1016/S0034-4877(97)89757-1.

R. J. Hughes, “Finite-rank perturbations of the Dirac operator,” J. Math. Anal. Appl., vol. 238, no. 1, pp. 67–81, 1999, doi: 10.1006/jmaa.1999.6504.

V. Lotoreichik and T. Ourmières-Bonafos, “Spectral asymptotics of the Dirac operator in a thin shell,” 2023, arXiv:2307.09033.

A. Mas and F. Pizzichillo, “The relativistic spherical δ-shell interaction in R3: spectrum and approximation,” J. Math. Phys., vol. 58, no. 8, 2017, Art. ID 082102, doi: 10.1063/1.5000381.

A. Mas and F. Pizzichillo, “Klein’s paradox and the relativistic δ-shell interaction in R3,” Anal. PDE, vol. 11, no. 3, pp. 705–744, 2018, doi: 10.2140/apde.2018.11.705.

M. Reed and B. Simon, Methods of modern mathematical physics. I Functional analysis, 2nd ed. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980.

P. Šeba, “Klein’s paradox and the relativistic point interaction,” Lett. Math. Phys., vol. 18, no. 1, pp. 77–86, 1989, doi: 10.1007/BF00397060.

B. Thaller, The Dirac equation, ser. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992, doi: 10.1007/978-3-662-02753-0.

J. A. Thorpe, Elementary topics in differential geometry, ser. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1979.

M. Tušek, “Approximation of one-dimensional relativistic point interactions by regular potentials revised,” Lett. Math. Phys., vol. 110, no. 10, pp. 2585–2601, 2020, doi: 10.1007/s11005- 020-01325-6.