zbMATH — the first resource for mathematics

Resolvent convergence to Dirac operators on planar domains. (English) Zbl 1414.81103
Summary: Consider a Dirac operator defined on the whole plane with a mass term of size \(m\) supported outside a domain \(\Omega \). We give a simple proof for the norm resolvent convergence, as \(m\) goes to infinity, of this operator to a Dirac operator defined on \(\Omega \) with infinite-mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
46N50 Applications of functional analysis in quantum physics
81Q37 Quantum dots, waveguides, ratchets, etc.
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
47A10 Spectrum, resolvent
Full Text: DOI
[1] Berry, MV; Mondragon, RJ, Neutrino billiards: time-reversal symmetry-breaking without magnetic fields, Proc. R. Soc. Lond. Ser. A, 412, 53-74, (1987)
[2] Stockmeyer, E.; Vugalter, S., Infinite mass boundary conditions for Dirac operators, J. Spectr. Theory, (2018)
[3] Neto, AHC; Guinea, F.; Peres, NMR; Novoselov, KS; Geim, AK, The electronic properties of graphene, Rev. Mod. Phys., 81, 109-162, (2009)
[4] Armitage, NP; Mele, EJ; Vishwanath, A., Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys., 90, 015001, (2018)
[5] Akhmerov, AR; Beenakker, CWJ, Boundary conditions for Dirac fermions on a terminated honeycomb lattice, Phys. Rev. B, 77, 085423, (2008)
[6] Hunt, B.; Sanchez-Yamagishi, JD; Young, AF; Yankowitz, M.; LeRoy, BJ; Watanabe, K.; Taniguchi, T.; Moon, P.; Koshino, M.; Jarillo-Herrero, P.; etal., Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure, Science, 340, 1427-1430, (2013)
[7] Lu, J., Watson, A.B., Weinstein, M.I.: Dirac operators and domain walls (2018). arXiv:1808.01378
[8] Barbaroux, J-M; Cornean, H.; Stockmeyer, E., Spectral gaps in graphene antidot lattices, Integr. Equ. Oper. Theory, 89, 631-646, (2017) · Zbl 1381.81044
[9] Raedt, H.; Katsnelson, MI, Electron energy level statistics in graphene quantum dots, JETP Lett., 88, 607-610, (2009)
[10] Brun, SJ; Pereira, VM; Pedersen, TG, Boron and nitrogen doping in graphene antidot lattices, Phys. Rev. B, 93, 245420, (2016)
[11] Brun, SJ; Thomsen, MR; Pedersen, TG, Electronic and optical properties of graphene antidot lattices: comparison of Dirac and tight-binding models, J. Phys.: Cond. Matter, 26, 265301, (2014)
[12] Leoni, G.: A First Course in Sobolev Spaces. American Mathematical Society, Providence (2009) · Zbl 1180.46001
[13] Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (2010) · Zbl 1194.35001
[14] Benguria, RD; Fournais, S.; Stockmeyer, E.; Bosch, H., Self-adjointness of two-dimensional Dirac operators on domains, Ann. Henri Poincaré, 18, 1371-1383, (2017) · Zbl 1364.81117
[15] Benguria, RD; Fournais, S.; Stockmeyer, E.; Bosch, H., Spectral gaps of Dirac operators describing graphene quantum dots, Math. Phys. Anal. Geom., 20, 11, (2017) · Zbl 1424.81011
[16] Borrelli, W., Multiple solutions for a self-consistent Dirac equation in two dimensions, J. Math. Phys., 59, 041503, (2018) · Zbl 1393.81013
[17] Treust, L.; Ourmières-Bonafos, T., Self-adjointness of Dirac operators with infinite mass boundary conditions in sectors, Ann. Henri Poincaré, 19, 1465-1487, (2018) · Zbl 1388.81108
[18] Bogolioubov, PN, Sur un modèle à quarks quasi-indépendants, Ann. l’I.H.P., Sect. A, 8, 163-189, (1968)
[19] Chodos, A.; Jaffe, RL; Johnson, K.; Thorn, CB, Baryon structure in the bag theory, Phys. Rev. D, 10, 2599-2604, (1974)
[20] Arrizabalaga, N., Le Treust, L., Mas, A., Raymond, N.: The MIT bag model as an infinite mass limit (2018). arXiv:1808.09746
[21] Arrizabalaga, N.; Treust, L.; Raymond, N., On the MIT bag model in the non-relativistic limit, Commun. Math. Phys., 354, 641-669, (2017) · Zbl 1373.81422
[22] Arrizabalaga, N., Le Treust, L., Raymond, N.: Extension operator for the MIT bag model. Ann. la Fac. Sci. Toulouse Math. (2018) · Zbl 1373.81422
[23] de Oliveira, C.R.: Intermediate Spectral Theory and Quantum Dynamics. Birkhauser, Basel (2008)
[24] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, study edn. Springer, Berlin (1987) · Zbl 0619.47005
[25] Weidmann, J.: Lineare Operatoren in Hilberträumen, Volume of Mathematische Leitfäden. Teubner, Stuttgart (1976) · Zbl 0344.47001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.