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

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     title = {The {MIT} {Bag} {Model} as an infinite mass limit},
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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/

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