The MIT Bag Model as an infinite mass limit

Arrizabalaga, Naiara; Le Treust, Loïc; Mas, Albert; Raymond, Nicolas

doi:10.5802/jep.95

The MIT Bag Model as an infinite mass limit
[Le modèle MIT bag obtenu comme une limite de masse grande]

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

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 329-365.

Résumé
Abstract

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 $Ω \subset ℝ^{3}$ . 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.

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 $Ω \subset ℝ^{3}$ , 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 $o (1 / \sqrt{m})$ , is carried out by introducing tubular coordinates in a neighborhood of $\partial Ω$ and analyzing one dimensional optimization problems in the normal direction.

Reçu le : 2018-09-12
Accepté le : 2019-05-08
Publié le : 2019-05-23

MR Zbl

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

Affiliations des auteurs :

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

@article{JEP_2019__6__329_0,
     author = {Arrizabalaga, Naiara and Le Treust, Lo{\"\i}c and Mas, Albert and Raymond, Nicolas},
     title = {The {MIT} {Bag} {Model} as an infinite mass limit},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {329--365},
     publisher = {Ecole polytechnique},
     volume = {6},
     year = {2019},
     doi = {10.5802/jep.95},
     zbl = {07070236},
     mrnumber = {3959076},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jep.95/}
}

TY  - JOUR
AU  - Arrizabalaga, Naiara
AU  - Le Treust, Loïc
AU  - Mas, Albert
AU  - Raymond, Nicolas
TI  - The MIT Bag Model as an infinite mass limit
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2019
SP  - 329
EP  - 365
VL  - 6
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.95/
DO  - 10.5802/jep.95
LA  - en
ID  - JEP_2019__6__329_0
ER  -

%0 Journal Article
%A Arrizabalaga, Naiara
%A Le Treust, Loïc
%A Mas, Albert
%A Raymond, Nicolas
%T The MIT Bag Model as an infinite mass limit
%J Journal de l’École polytechnique — Mathématiques
%D 2019
%P 329-365
%V 6
%I Ecole polytechnique
%U http://www.numdam.org/articles/10.5802/jep.95/
%R 10.5802/jep.95
%G en
%F JEP_2019__6__329_0

Arrizabalaga, Naiara; Le Treust, Loïc; Mas, Albert; Raymond, Nicolas. The MIT Bag Model as an infinite mass limit. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 329-365. doi : 10.5802/jep.95. http://www.numdam.org/articles/10.5802/jep.95/

Bibliographie
Cité par

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

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

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

[4] Bär, Christian; Ballmann, Werner Boundary value problems for elliptic differential operators of first order (Surv. Differ. Geom.), Volume XVII, Int. Press, Boston, MA, 2012, pp. 1-78 | Zbl

[5] Barbaroux, L. J. Cornean H. D. Le Treust; Stockmeyer, E. 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] Benguria, R. D.; Fournais, S.; Stockmeyer, E.; Van Den Bosch, H. Spectral gaps of Dirac operators with boundary conditions relevant for graphene (2016) (arXiv:1601.06607)

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

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

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

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

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

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

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

[14] Mas, Albert; Pizzichillo, Fabio Klein’s paradox and the relativistic $δ$ -shell interaction in $ℝ^{3}$ , Anal. PDE, Volume 11 (2018) no. 3, pp. 705-744 | MR | Zbl

[15] Ourmières-Bonafos, Thomas; Vega, Luis A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and $δ$ -shell interactions, Publ. Mat., Volume 62 (2018) no. 2, pp. 397-437 | DOI | MR | Zbl

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

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

Cité par Sources :