Geodesic
Constructive Semiclassical Asymptotics of Bound States of Graphene in a Constant Magnetic Field with Small Mass
Matematičeskie zametki, Tome 111 (2022) no. 2, pp. 163-187.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with constructive semiclassical asymptotics of eigenfunctions of the Dirac operator describing graphene in a constant magnetic field. Two cases are considered: (a) a strong magnetic field, and (b) a radially symmetric electric field and small mass. Using standard semiclassical methods, we reduce the problem to a pencil of magnetic Schrödinger operators with a correction term. In both cases, the classical system defined by the principal symbol turns out to be integrable, but the correction term destroys the integrability. In case (a), where the correction removes the frequency degeneracy (resonance), we use the averaging method to reduce the problem to an integrable system not only in the leading approximation but also with the correction taken into account. The tori of the resulting system generate a series of asymptotic eigenfunctions of the original operator. In case (b), the system defined by the principal symbol is nondegenerate. Fixing an invariant torus with Diophantine frequencies for this system and looking for a solution of the transport equation for it, we obtain a series of asymptotic eigenfunctions that are in one-to-one correspondence with tori that satisfy the Bohr–Sommerfeld quantization rule and lie in a small neighborhood of the chosen Diophantine torus. In both cases, the construction of the asymptotic eigenfunctions is based on the global representation of the Maslov canonical operator on a two-dimensional torus projected onto the configuration space into an annular domain with two simple caustics via the Airy function and its derivative. A numerical implementation of our formulas in examples shows their efficiency.
Keywords: semiclassical asymptotics, Dirac operator, graphene in a magnetic field, averaging, Airy function. 
@article{MZM_2022_111_2_a0,
     author = {A. Yu. Anikin and V. V. Rykhlov},
     title = {Constructive {Semiclassical} {Asymptotics} of {Bound} {States} of {Graphene} in a {Constant} {Magnetic} {Field} with {Small} {Mass}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {163--187},
     publisher = {mathdoc},
     volume = {111},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_2_a0/}
}
TY  - JOUR
AU  - A. Yu. Anikin
AU  - V. V. Rykhlov
TI  - Constructive Semiclassical Asymptotics of Bound States of Graphene in a Constant Magnetic Field with Small Mass
JO  - Matematičeskie zametki
PY  - 2022
SP  - 163
EP  - 187
VL  - 111
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_111_2_a0/
LA  - ru
ID  - MZM_2022_111_2_a0
ER  - 
%0 Journal Article
%A A. Yu. Anikin
%A V. V. Rykhlov
%T Constructive Semiclassical Asymptotics of Bound States of Graphene in a Constant Magnetic Field with Small Mass
%J Matematičeskie zametki
%D 2022
%P 163-187
%V 111
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_111_2_a0/
%G ru
%F MZM_2022_111_2_a0
A. Yu. Anikin; V. V. Rykhlov. Constructive Semiclassical Asymptotics of Bound States of Graphene in a Constant Magnetic Field with Small Mass. Matematičeskie zametki, Tome 111 (2022) no. 2, pp. 163-187. http://geodesic.mathdoc.fr/item/MZM_2022_111_2_a0/

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

[2] V. P. Maslov, M. V. Fedoryuk, Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR | Zbl

[3] V. F. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions, Springer-Verlag, Berlin, 1993 | MR

[4] Y. Colin de Verdiére, “Spectre conjoint d'operateurs pseudo-differentiels qui commutent I. Le cas non integrable”, Duke Math. J., 46:1 (1979), 169–182 | DOI | Zbl

[5] A. Yu. Anikin, S. Yu. Dobrokhotov, “Diophantine tori and pragmatic calculation of quasimodes for operators with integrable principal symbol”, Russ. J. Math. Phys, 27:3 (2020), 299–308 | DOI | Zbl

[6] A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Ravnomernaya asimptotika v vide funktsii Eiri dlya kvaziklassicheskikh svyazannykh sostoyanii v odnomernykh i radialno-simmetrichnykh zadachakh”, TMF, 201:3 (2019), 382–414 | DOI | MR | Zbl

[7] S. Yu. Dobrokhotov, V. E Nazaikinskii, “Efficient formulas for the Maslov canonical operator near a simple caustic”, Russ. J. Math. Phys., 25 (2018), 545–552 | DOI | Zbl

[8] S. Yu. Slavyanov, Asimptotika reshenii odnomernogo uravneniya Shredingera, Izd-vo LGU, L., 1990

[9] M. V. Berry, C. J. Howls, Integrals With Coalescing Saddles, NIST Digital Library of Mathematical Functions, 2010

[10] M. V. Fedoryuk, Asimptotika: integraly i ryady, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1987 | MR | Zbl

[11] S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Lagranzhevy mnogoobraziya i effektivnye formuly dlya korotkovolnovykh asimptotik v okrestnosti tochki vozvrata kaustiki”, Matem. zametki, 108:3 (2020), 334–359 | DOI | MR | Zbl

[12] N. N. Bogolyubov, Yu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR | Zbl

[13] R. G. Littlejohn, Hamilton Theory of Guiding Center Motion, Ph.D. Thesis, Berkeley, CA, 1980

[14] A. I. Neishtadt, “Razdelenie dvizhenii v sistemakh s bystro vraschayuscheisya fazoi”, PMM, 48:2 (1984), 197–204

[15] M. V. Karasev, V. P. Maslov, “Asimptoticheskoe i geometricheskoe kvantovanie”, UMN, 39:6 (1984), 115–173 | MR | Zbl

[16] M. V. Karasev, E. M. Novikova, “Algebras with polynomial commutation relations for a quantum particle in electric and magnetic fields”, Quantum algebras and Poisson geometry in mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 216, Amer. Math. Soc., Providence, RI, 2005, 19–135 | MR | Zbl

[17] W. Scherer, “Quantum Averaging. I. Poincaré von Zeipel is Rayleigh Schrödinger”, J. Phys. A: Math. Gen., 27:24 (1994), 8231 | DOI | Zbl

[18] W. Scherer, “Quantum Averaging. II. Kolmogorov's Algorithm”, J. Phys. A: Math. Gen., 30:8 (1997), 2825 | DOI | Zbl

[19] J. Brüning, S. Yu. Dobrokhotov, K. V. Pankrashkin, “The spectral asymptotics of the two-dimensional Schrödinger operator with a strong magnetic field. I”, Russ. J. Math. Phys., 9:1 (2002), 14–49 | MR | Zbl

[20] J. Brüning, S. Yu. Dobrokhotov, K. V. Pankrashkin, “The spectral asymptotics of the two-dimensional Schrödinger operator with a strong magnetic field. II”, Russ. J. Math. Phys., 9:4 (2002), 400–416 | MR | Zbl

[21] A. Yu. Anikin, J. Brüning, S. Yu. Dobrokhotov, E. Vybornyi, “Averaging and spectral bands for 2-D magnetic Schrödinger operator with the growing and one-dimension periodic potential”, Russ. J. Math. Phys., 26:3 (2019), 265–276 | DOI | Zbl

[22] A. Eckstein, “Unitary reduction for the two-dimensional Schrodinger operator with strong magnetic field”, Math. Nachr., 282:4 (2009), 504–525 | DOI | Zbl

[23] B. Helffer, J. Sjöstrand, “Équation de Schrödinger avec champ magnétique et équation de Harper”, Schrödinger Operators, Lecture Notes in Phys., 345, Springer, Berlin, 1998, 118–198 | DOI

[24] A. Yu. Anikin, S. Yu. Dobrokhotov, A. I. Klevin, B. Tirotstsi, “Skalyarizatsiya statsionarnykh kvaziklassicheskikh zadach dlya sistem uravnenii i prilozhenie k fizike plazmy”, TMF, 193:3 (2017), 409–433 | DOI | Zbl

[25] S. Yu. Dobrokhotov, D. S. Minenkov, M. Rulo, “Printsip Mopertyui–Yakobi dlya gamiltonianov vida $f(x,|p|)$ v nekotorykh dvumernykh statsionarnykh kvaziklassicheskikh zadachakh”, Matem. zametki, 97:1 (2015), 48–57 | DOI | MR | Zbl

[26] M. V. Karasev, E. M. Novikova, “Predstavlenie tochnykh i kvaziklassicheskikh sobstvennykh funktsii cherez kogerentnye sostoyaniya. Atom vodoroda v magnitnom pole”, TMF, 108:3 (1996), 339–387 | DOI | MR | Zbl

[27] V. I. Arnold, A. N. Varchenko, S. M. Gusein-Zade, Osobennosti differentsiruemykh otobrazhenii. Tom 1. Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982 | MR