×

On the MIT bag model in the non-relativistic limit. (English) Zbl 1373.81422

Summary: This paper is devoted to the spectral investigation of the MIT bag model, that is, the Dirac operator on a smooth and bounded domain of \({\mathbb{R}^3}\) with certain boundary conditions. When the mass \(m\) goes to \({\pm\infty}\), we provide spectral asymptotic results.

MSC:

81V35 Nuclear physics
81V05 Strong interaction, including quantum chromodynamics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35F15 Boundary value problems for linear first-order PDEs
35B40 Asymptotic behavior of solutions to PDEs

References:

[1] Arrizabalaga N., Mas A., Vega L.: Shell interactions for Dirac operators: on the point spectrum and the confinement. SIAM J. Math. Anal. 47(2), 1044-1069 (2015) · Zbl 1314.81083 · doi:10.1137/14097759X
[2] Bär, C., Ballmann, W.: Boundary value problems for elliptic differential operators of first order. In: Surveys in Differential Geometry, Vol. 17 of Surv. Differ. Geom., pp. 1-78. Int. Press, Boston, MA (2012) · Zbl 1331.58022
[3] Bär C., Ballmann W.: Guide to Elliptic Boundary Value Problems for Dirac-Type Operators, pp. 43-80. Springer International Publishing, Cham (2016) · Zbl 1377.58022
[4] Benguria R.D., Fournais S., Stockmeyer E., Van Den Bosch H.: Spectral gaps of Dirac operators with boundary conditions relevant for graphene. Math. Phys. Anal. Geom. 20, 11 (2017) · Zbl 1424.81011 · doi:10.1007/s11040-017-9242-4
[5] Bogolioubov P.: Sur un modèle à quarks quasi-indépendants. Ann. l’I.H.P. Sect. A 8, 163-189 (1968)
[6] Booß-Bavnbek B., Lesch M., Zhu C.: The Calderón projection: new definition and applications. J. Geom. Phys. 59(7), 784-826 (2009) · Zbl 1221.58016 · doi:10.1016/j.geomphys.2009.03.012
[7] Chodos A.: Field-theoretic Lagrangian with baglike solutions. Phys. Rev. D (3) 12(8), 2397-2406 (1975) · doi:10.1103/PhysRevD.12.2397
[8] Chodos A., Jaffe R.L., Johnson K., Thorn C.B.: Baryon structure in the bag theory. Phys. Rev. D 10, 2599-2604 (1974) · doi:10.1103/PhysRevD.10.2599
[9] Chodos A., Jaffe R.L., Johnson K., Thorn C.B., Weisskopf V.F.: New extended model of hadrons. Phys. Rev. D (3) 9(12), 3471-3495 (1974) · Zbl 1170.34030 · doi:10.1103/PhysRevD.9.3471
[10] DeGrand T., Jaffe R.L., Johnson K., Kiskis J.: Masses and other parameters of the light hadrons. Phys. Rev. D 12, 2060-2076 (1975) · doi:10.1103/PhysRevD.12.2060
[11] Dittrich J., Exner P., Kühn C., Pankrashkin K.: On eigenvalue asymptotics for strong \[{\delta}\] δ-interactions supported by surfaces with boundaries. Asymptot. Anal. 97(1-2), 1-25 (2016) · Zbl 1398.35138 · doi:10.3233/ASY-151341
[12] Griffiths D.: Introduction to Elementary Particles. Wiley, New York (2008) · Zbl 1162.00012
[13] Helffer B.: Semi-classical Analysis for the Schrödinger Operator and Applications, vol. 1336 of Lecture Notes in Mathematics. Springer, Berlin (1988) · Zbl 0647.35002
[14] Helffer B., Kachmar A.: Eigenvalues for the Robin Laplacian in domains with variable curvature. Trans. Am. Math. Soc. 369(5), 3253-3287 (2017) · Zbl 1364.35215 · doi:10.1090/tran/6743
[15] Helffer, B., Kachmar, A., Raymond, N.: Tunneling for the Robin Laplacian in smooth planar domains. Commun. Contemp. Math. 19(1), 1650030 (2017) · Zbl 1361.35119
[16] Hijazi O., Montiel S., Roldán A.: Eigenvalue boundary problems for the Dirac operator. Commun. Math. Phys. 231(3), 375-390 (2002) · Zbl 1018.58020 · doi:10.1007/s00220-002-0725-0
[17] Hosaka A., Toki H.: Quarks, Baryons and Chiral Symmetry. World Scientific, Singapore (2001) · doi:10.1142/4708
[18] Jecko, T.: On the mathematical treatment of the Born-Oppenheimer approximation. J. Math. Phys. 55(5), 053504 (2014) · Zbl 1298.81092
[19] Johnson K.: The MIT bag model. Acta Phys. Pol. B(6), 865-892 (1975)
[20] Kachmar, A., Keraval, P., Raymond, N.: Weyl formulae for the Robin Laplacian in the semiclassical limit. Conflu. Math. arXiv:1602.06179 (2017) · Zbl 1516.35282
[21] Kato T.: Perturbation Theory for Linear Operators. Die Grundlehren der Mathematischen Wissenschaften, Band 132. Springer, New York (1981)
[22] Lampart, J., Teufel, S.: The adiabatic limit of the Laplacian on thin fibre bundles. In: Microlocal Methods in Mathematical Physics and Global Analysis, Trends Math., pp. 33-36. Birkhäuser/Springer, Basel (2013) · Zbl 1264.58017
[23] Le Treust, L.: Variational and topological methods for the study of nonlinear models from relativistic quantum mechanics. Theses, Université Paris Dauphine, Paris IX (2013)
[24] Messiah, A.: Quantum Mechanics. Vol. I. Translated from the French by G. M. Temmer. North-Holland Publishing Co., Amsterdam; Interscience Publishers Inc., New York (1961) · Zbl 0102.42602
[25] Ourmières-Bonafos, T., Vega, L.: A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and shell interactions. Preprint (2017) · Zbl 06918953
[26] Pankrashkin K., Popoff N.: An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter. J. Math. Pures Appl. (9) 106(4), 615-650 (2016) · Zbl 1345.35068 · doi:10.1016/j.matpur.2016.03.005
[27] Raymond, N.: Bound States of the Magnetic Schrödinger Operator. Number 27. EMS Tracts (2017) · Zbl 1370.35004
[28] Reiher M., Wolf A.: Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science. Wiley, New York (2014) · doi:10.1002/9783527667550
[29] Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. III, 2nd edn. Publish or Perish, Inc., Wilmington (1979) · Zbl 0439.53002
[30] Stockmeyer, E., Vugalter, S.: Infinite mass boundary conditions for Dirac operators. arXiv preprint arXiv:1603.09657 (2016) · Zbl 1419.81014
[31] Thaller B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)
[32] Thomas, A.W.: Advances in Nuclear Physics: Volume 13, Chapter Chiral Symmetry and the Bag Model: A New Starting Point for Nuclear Physics, pp. 1-137. Springer, Boston (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.