Geodesic
Fundamental solution of the stationary Dirac equation with a linear
Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 3, pp. 349-367 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier transforms of parabolic cylinder functions. This solution describes the flux of quasiparticles in graphene emitted by a pointlike source of electrons that are partially converted into holes (antiparticles). Using our explicit formula, we calculate its semiclassical asymptotic behavior in the hole region.
Keywords: massless Dirac equation, semiclassical asymptotic behavior, fundamental solution, Green's matrix, parabolic cylinder function
Mots-clés : graphene, quasiparticle. 
@article{TMF_2020_205_3_a0,
     author = {I. Bogaevsky},
     title = {Fundamental solution of the~stationary {Dirac} equation with a~linear},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {349--367},
     year = {2020},
     volume = {205},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a0/}
}
TY  - JOUR
AU  - I. Bogaevsky
TI  - Fundamental solution of the stationary Dirac equation with a linear
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2020
SP  - 349
EP  - 367
VL  - 205
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a0/
LA  - ru
ID  - TMF_2020_205_3_a0
ER  - 
%0 Journal Article
%A I. Bogaevsky
%T Fundamental solution of the stationary Dirac equation with a linear
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2020
%P 349-367
%V 205
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a0/
%G ru
%F TMF_2020_205_3_a0
I. Bogaevsky. Fundamental solution of the stationary Dirac equation with a linear. Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 3, pp. 349-367. http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a0/

[1] M. I. Katsnelson, Graphene: Carbon in Two Dimensions, Cambridge Univ. Press, Cambridge, 2012

[2] T. Tudorovskiy, K. J. A. Reijnders, M. I. Katsnelson, “Chiral tunneling in single-layer and bilayer graphene”, Phys. Scr., 2012:T146 (2012), 014010, 19 pp. | DOI

[3] S. Y. Dobrokhotov, A. A. Tolchennikov, “Solution of the two-dimensional Dirac equation with a linear potential and a localized initial condition”, Russ. J. Math. Phys., 26:2 (2019), 139–151 | DOI | MR

[4] P. Carmier, D. Ullmo, “Berry phase in graphene: Semiclassical perspective”, Phys. Rev. B, 77:24 (2008), 245413, arXiv: 0801.4727 | DOI

[5] J. B. Keller, “Geometrical theory of diffraction”, J. Opt. Soc. Amer., 52:2 (1962), 116–130 | DOI | MR

[6] V. M. Babich, “O korotkovolnovoi asimptotike funktsii Grina dlya uravneniya Gelmgoltsa”, Matem. sb., 65(107):4 (1964), 576–630 | MR | Zbl

[7] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[8] V. V. Kucherenko, “Korotkovolnovaya asimptotika funktsii Grina dlya $N$-mernogo volnovogo uravneniya v neodnorodnoi srede”, Zhurn. vychisl. matem. matem. fiz., 8:4 (1968), 908–913 | DOI | MR | Zbl

[9] A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rulo, “Kanonicheskii operator Maslova na pare lagranzhevykh mnogoobrazii i asimptotika reshenii statsionarnykh uravnenii s lokalizovannymi pravymi chastyami”, Dokl. RAN, 475:6 (2017), 624–628 | DOI

[10] V. I. Arnold, Osobennosti kaustik i volnovykh frontov, Fazis, M., 1996 | DOI

[11] V. I. Arnold, “On the interior scattering of waves, defined by hyperbolic variational principles”, J. Geom. Phys., 5:3 (1988), 305–315 | DOI | MR

[12] I. A. Bogaevskii, “Kaustiki vnutrennego rasseyaniya”, Tr. MIAN, 267 (2009), 7–13 | DOI | MR | Zbl

[13] I. A. Bogaevskii, “Perestroiki frontov vnutrennego rasseyaniya”, Dokl. RAN, 436:2 (2011), 155–158

[14] A. G. Sveshnikov, A. N. Bogolyubov, V. V. Kravtsov, Lektsii po matematicheskoi fizike, Izd-vo Mosk. un-ta, M., 1993 | MR

[15] A. Boutet de Monvel-Berthier, D. Manda, R. Purice, “Limiting absorption principle for the Dirac operator”, Ann. Inst. H. Poincaré Phys. Théor., 58:4 (1993), 413–431 | MR | Zbl

[16] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 2, Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1974 | MR