Self-adjointness of two-dimensional Dirac operators on domains. (English) Zbl 1364.81117
Summary: We consider Dirac operators defined on planar domains. For a large class of boundary conditions, we give a direct proof of their self-adjointness in the Sobolev space \(H^1\).
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
References:
| [1] | Adams, R.A.: Sobolev Spaces, Volume 65 of Pure and Applied Mathematics. Academic Press, New York (1975) · Zbl 0314.46030 |
| [2] | Akhmerov, A.R., Beenakker, C.W.J.: Boundary conditions for Dirac fermions on a terminated honeycomb lattice. Phys. Rev. B 77, 085423 (2008) · doi:10.1103/PhysRevB.77.085423 |
| [3] | Beneventano, C.G., Fialkovsky, I., Santangelo, E.M., Vassilevich, D.V.: Charge density and conductivity of disordered Berry-Mondragon graphene nanoribbons. Eur. Phys. J. B 87(3), 1-9 (2014) · doi:10.1140/epjb/e2014-40990-x |
| [4] | Berry, M.V., Mondragon, R.J.: Neutrino billiards: time-reversal symmetry-breaking without magnetic fields. Proc. R. Soc. Lond. Ser. A 412(1842), 53-74 (1987) · doi:10.1098/rspa.1987.0080 |
| [5] | Booß-Bavnbek, B., Lesch, M., Zhu, Ch.: 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 |
| [6] | Booß-Bavnbek, B., Wojciechowski, K.P.: Elliptic boundary problems for Dirac operators. In: Mathematics Theory & Applications. Birkhäuser Boston Inc, Boston (1993) · Zbl 0797.58004 |
| [7] | Brezis, H.: Analyse fonctionnelle. In: Collection Mathématiques Appliquées pour la Maîtrise. Théorie et applications. [Theory and applications]. Masson, Paris (1983) · Zbl 0511.46001 |
| [8] | 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) · doi:10.1103/RevModPhys.81.109 |
| [9] | Freitas, P., Siegl, P.: Spectra of graphene nanoribbons with armchair and zigzag boundary conditions. Rev. Math. Phys. 26(10), 1450018 (2014) · Zbl 1309.47088 · doi:10.1142/S0129055X14500184 |
| [10] | Pommerenke, Ch.: Boundary behaviour of conformal maps. In: Grundlehren der mathematischen Wissenschaften [A Series of Comprehensive Studies in Mathematics]. Springer, Berlin (1991) · Zbl 0491.30009 |
| [11] | Prokhorova, M.: The spectral flow for Dirac operators on compact planar domains with local boundary conditions. Commun. Math. Phys. 322(2), 385-414 (2013) · Zbl 1281.58015 · doi:10.1007/s00220-013-1701-6 |
| [12] | Schmidt, K.M.: A remark on boundary value problems for the Dirac operator. Q. J. Math. Oxf. Ser. (2) 46(184), 509-516 (1995) · Zbl 0864.35092 · doi:10.1093/qmath/46.4.509 |
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.
