The MIT Bag Model as an infinite mass limit
Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 329-365.

The Dirac operator, acting in three dimensions, is considered. Assuming that a large mass m>0 lies outside a smooth enough and bounded open set Ω 3 , it is proved that its spectrum approximates the one of the Dirac operator on Ω with the MIT bag boundary condition. The approximation, modulo an error of order o(1/m), is carried out by introducing tubular coordinates in a neighborhood of Ω and analyzing one dimensional optimization problems in the normal direction.

Nous considérons l’opérateur de Dirac en dimension 3 dont la masse m>0 est supposée grande à l’extérieur d’un ouvert borné et régulier Ω 3 . Nous démontrons que son spectre approche celui de l’opérateur de Dirac sur Ω qui intègre dans son domaine les conditions au bord dites « MIT bag ». L’analyse asymptotique est réalisée grâce à l’usage de coordonnées tubulaires et à l’analyse d’un problème d’optimisation unidimensionnel dans la direction normale.

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DOI: 10.5802/jep.95
Classification: 35J60, 35Q75, 49J45, 49S05, 81Q10, 81V05, 35P15, 58C40
Keywords: Dirac operator, relativistic particle in a box, MIT bag model, spectral theory
Mot clés : Opérateur de Dirac, particules relativistes dans une boîte, modèle MIT bag, théorie spectrale

Naiara Arrizabalaga 1; Loïc Le Treust 2; Albert Mas 3; Nicolas Raymond 4

1 Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU) 48080 Bilbao, Spain
2 Aix Marseille Univ, CNRS, Centrale Marseille, I2M Marseille, France
3 Departament de Matemàtiques, Universitat Politècnica de Catalunya Campus Diagonal Besòs, Edifici A (EEBE), Av. Eduard Maristany 16, 08019 Barcelona, Spain
4 Laboratoire Angevin de Recherche en Mathématiques, LAREMA, UMR 6093, UNIV Angers, SFR Math-STIC 2, boulevard Lavoisier, 49045 Angers Cedex 01, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Naiara Arrizabalaga; Loïc Le Treust; Albert Mas; Nicolas Raymond. The MIT Bag Model as an infinite mass limit. Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 329-365. doi : 10.5802/jep.95. https://jep.centre-mersenne.org/articles/10.5802/jep.95/

[1] A. R. Akhmerov & C. W. J. Beenakker - “Boundary conditions for Dirac fermions on a terminated honeycomb lattice”, Phys. Rev. B 77 (2008) no. 8, article # 085423 | DOI

[2] N. Arrizabalaga, L. Le Treust & N. Raymond - “On the MIT bag model in the non-relativistic limit”, Comm. Math. Phys. 354 (2017) no. 2, p. 641-669 | DOI | MR | Zbl

[3] N. Arrizabalaga, L. Le Treust & N. Raymond - “Extension operator for the MIT bag model”, Ann. Fac. Sci. Toulouse Math. (6) (2019), to appear | Zbl

[4] C. Bär & W. Ballmann - “Boundary value problems for elliptic differential operators of first order”, Surv. Differ. Geom., vol. XVII, Int. Press, Boston, MA, 2012, p. 1-78 | Zbl

[5] L. Barbaroux & E. Stockmeyer - “Resolvent convergence to Dirac operators on planar domains”, Ann. Henri Poincaré (2019), doi:10.1007/s00023-019-00787-2, arXiv:1810.02957 | DOI | MR | Zbl

[6] R. D. Benguria, S. Fournais, E. Stockmeyer & H. Van Den Bosch - “Spectral gaps of Dirac operators with boundary conditions relevant for graphene” (2016), arXiv:1601.06607

[7] M. V. Berry & R. J. Mondragon - “Neutrino billiards: time-reversal symmetry-breaking without magnetic fields”, Proc. Roy. Soc. London Ser. A 412 (1987) no. 1842, p. 53-74 | DOI | MR

[8] P. Bogolioubov - “Sur un modèle à quarks quasi-indépendants”, Ann. Inst. H. Poincaré Sect. A 8 (1968), p. 163-189

[9] B. Booß-Bavnbek, M. Lesch & C. Zhu - “The Calderón projection: new definition and applications”, J. Geom. Phys. 59 (2009) no. 7, p. 784-826 | DOI | Zbl

[10] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn & V. F. Weisskopf - “New extended model of hadrons”, Phys. Rev. D (3) 9 (1974) no. 12, p. 3471-3495 | MR

[11] T. DeGrand, R. L. Jaffe, K. Johnson & J. Kiskis - “Masses and other parameters of the light hadrons”, Phys. Rev. D 12 (1975) no. 7, p. 2060-2076

[12] L. C. Evans - Partial differential equations, Graduate studies in Math., vol. 19, American Mathematical Society, Providence, RI, 2010 | Zbl

[13] K. Johnson - “The MIT bag model”, Acta Phys. Polon. B 6 (1975), p. 865-892

[14] A. Mas & F. Pizzichillo - “Klein’s paradox and the relativistic δ-shell interaction in 3 , Anal. PDE 11 (2018) no. 3, p. 705-744 | MR | Zbl

[15] T. Ourmières-Bonafos & L. Vega - “A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions”, Publ. Mat. 62 (2018) no. 2, p. 397-437 | DOI | MR | Zbl

[16] E. Stockmeyer & S. Vugalter - “Infinite mass boundary conditions for Dirac operators”, J. Spectral Theory 9 (2019) no. 2, p. 569-600 | DOI | MR | Zbl

[17] B. Thaller - The Dirac equation, Theoretical and Mathematical Physics, Springer-Verlag, Berlin, 1992 | MR | Zbl

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