Abstract
This paper deals with the study of the two-dimensional Dirac operator with infinite mass boundary conditions in sectors. We investigate the question of self-adjointness depending on the aperture of the sector: when the sector is convex it is self-adjoint on a usual Sobolev space, whereas when the sector is non-convex it has a family of self-adjoint extensions parametrized by a complex number of the unit circle. As a by-product of the analysis, we are able to give self-adjointness results on polygonal domains. We also discuss the question of distinguished self-adjoint extensions and study basic spectral properties of the Dirac operator with a mass term in the sector.
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Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, vol. 55. Courier Corporation, North Chelmsford (1964)
Akhmerov, A.R., Beenakker, C.W.J.: Boundary conditions for Dirac fermions on a terminated honeycomb lattice. Phys. Rev. B 77, 085423 (2008)
Akola, J., Heiskanen, H.P., Manninen, M.: Edge-dependent selection rules in magic triangular graphene flakes. Phys. Rev. B 77, 193410 (2008)
Arrizabalaga, N., Le Treust, L., Raymond, N.: Extension operator for the MIT bag model. Ann. Fac. Sci. Toulouse Math. (2017) (to appear)
Arrizabalaga, N., Le Treust, L., Raymond, N.: On the MIT bag model in the non-relativistic limit. Commun. Math. Phys. 354(2), 641–669 (2017)
Bär, C., Ballmann, W.: Guide to elliptic boundary value problems for Dirac-type operators. In: Arbeitstagung Bonn 2013, Volume 319 of Progress in Mathematics, pp. 43–80. Birkhäuser/Springer, Cham (2016)
Benguria, R.D., Fournais, S., Stockmeyer, E., Van Den Bosch, H.: Self-adjointness of two-dimensional Dirac operators on domains. Ann. Henri Poincaré 18(4), 1371–1383 (2017)
Booß Bavnbek, B., Lesch, M., Zhu, C.: The Calderón projection: new definition and applications. J. Geom. Phys. 59(7), 784–826 (2009)
Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)
Chodos, A.: Field-theoretic Lagrangian with baglike solutions. Phys. Rev. D 12(8), 2397–2406 (1975)
Chodos, A., Jaffe, R.L., Johnson, K., Thorn, C.B.: Baryon structure in the bag theory. Phys. Rev. D 10(8), 2599–2604 (1974)
Chodos, A., Jaffe, R.L., Johnson, K., Thorn, C.B., Weisskopf, V.F.: New extended model of hadrons. Phys. Rev. D 9(12), 3471–3495 (1974)
DeGrand, T., Jaffe, R.L., Johnson, K., Kiskis, J.: Masses and other parameters of the light hadrons. Phys. Rev. D 12, 2060–2076 (1975)
Demengel, F., Demengel, G.: Espaces fonctionnels. Savoirs Actuels (Les Ulis) [Current Scholarship (Les Ulis)]. EDP Sciences, Les Ulis; CNRS Éditions, Paris (2007). Utilisation dans la résolution des équations aux dérivées partielles [Application to the solution of partial differential equations]
Dittrich, J., Exner, P., Šeba, P.: Dirac operators with a spherically symmetric \(\delta \)-shell interaction. J. Math. Phys. 30(12), 2875–2882 (1989)
Esteban, M.J., Lewin, M., Séré, E.: Domains for Dirac-Coulomb min–max levels. Rev. Mat. Iberoam. (2018) (to appear)
Freitas, P., Siegl, P.: Spectra of graphene nanoribbons with armchair and zigzag boundary conditions. Rev. Math. Phys. 26(10), 1450018 (2014)
Grisvard, P.: Boundary Value Problems in Non-smooth Domains, vol. 19. Department of Mathematics, University of Maryland, College Park (1980)
Grisvard, P.: Singularities in Boundary Value Problems. Volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Springer, Masson, Berlin, Paris (1992)
Hosaka, A., Toki, H.: Quarks, Baryons and Chiral Symmetry. World Scientific Publishing, Singapore (2001)
Johnson, K.: The MIT bag model. Acta Phys. Pol. B 6, 865–892 (1975)
Kim, S.C., Park, P.S., Yang, S.-R.E.: States near dirac points of a rectangular graphene dot in a magnetic field. Phys. Rev. B 81, 085432 (2010)
Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16, 209–292 (1967)
Necas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, New York (2011)
Olver, F.W.: Asymptotics and Special Functions. Academic Press, New York (2014)
Ourmières-Bonafos, T., Vega, L.: A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and delta-shell interactions. Publ. Mat. (2018) (to appear)
Potasz, P., Güçlü, A.D., Hawrylak, P.: Zero-energy states in triangular and trapezoidal graphene structures. Phys. Rev. B 81, 033403 (2010)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 2: Fourier Analysis, Self-Adjointness. Academic Press, New York (1978)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 4: Analysis of Operators. Academic Press, New York (1978)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 2: Fourier Analysis, Self-Adjointness. Gulf Professional Publishing, Oxford (1980)
Schmidt, K .M.: A remark on boundary value problems for the Dirac operator. Q. J. Math. Oxf. Ser. (2) 46(184), 509–516 (1995)
Schnez, S., Ensslin, K., Sigrist, M., Ihn, T.: Analytic model of the energy spectrum of a graphene quantum dot in a perpendicular magnetic field. Phys. Rev. B 78, 195427 (2008)
Stockmeyer, E., Vugalter, S.: Infinite mass boundary conditions for Dirac operators. J. Spectr. Theory (2018) (to appear)
Tang, C., Yan, W., Zheng, Y., Li, G., Li, L.: Dirac equation description of the electronic states and magnetic properties of a square graphene quantum dot. Nanotechnology 19(43), 435401 (2008)
Thaller, B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1991)
Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, vol. 1258. Springer, Berlin (1987)
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Communicated by Jan Derezinski.
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Le Treust, L., Ourmières-Bonafos, T. Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors. Ann. Henri Poincaré 19, 1465–1487 (2018). https://doi.org/10.1007/s00023-018-0661-y
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DOI: https://doi.org/10.1007/s00023-018-0661-y