Geodesic
Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 141 (2004) no. 2, pp. 267-303 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider equations of nonrelativistic quantum mechanics in thin three-dimensional tubes (nanotubes). We suggest a version of the adiabatic approximation that permits reducing the original three-dimensional equations to one-dimensional equations for a wide range of energies of longitudinal motion. The suggested reduction method (the operator method for separating the variables) is based on the Maslov operator method. We classify the solutions of the reduced one-dimensional equation. In Part I of this paper, we deal with the reduction problem, consider the main ideas of the operator separation of variables (in the adiabatic approximation), and derive the reduced equations. In Part II, we will discuss various asymptotic solutions and several effects described by these solutions.
Mots-clés : nanotubes, size quantization
Keywords: adiabatic approximation, spin-orbit interaction, semiclassical approximation. 
@article{TMF_2004_141_2_a6,
     author = {V. V. Belov and S. Yu. Dobrokhotov and T. Ya. Tudorovskii},
     title = {Asymptotic {Solutions} of {Nonrelativistic} {Equations} of {Quantum} {Mechanics} in {Curved} {Nanotubes:} {I.~Reduction} {to~Spatially} {One-Dimensional} {Equations}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {267--303},
     year = {2004},
     volume = {141},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2004_141_2_a6/}
}
TY  - JOUR
AU  - V. V. Belov
AU  - S. Yu. Dobrokhotov
AU  - T. Ya. Tudorovskii
TI  - Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2004
SP  - 267
EP  - 303
VL  - 141
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2004_141_2_a6/
LA  - ru
ID  - TMF_2004_141_2_a6
ER  - 
%0 Journal Article
%A V. V. Belov
%A S. Yu. Dobrokhotov
%A T. Ya. Tudorovskii
%T Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2004
%P 267-303
%V 141
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2004_141_2_a6/
%G ru
%F TMF_2004_141_2_a6
V. V. Belov; S. Yu. Dobrokhotov; T. Ya. Tudorovskii. Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 141 (2004) no. 2, pp. 267-303. http://geodesic.mathdoc.fr/item/TMF_2004_141_2_a6/

[1] R. Saito, G. Dresselhaus, M. S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998; M. S. Dresselhaus, G. Dresselhaus, P. C. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, 1996; И. В. Станкевич, М. В. Никеров, Д. А. Бочар, Усп. химии, 53:7 (1984), 640

[2] S. Iijima, Nature (London), 354 (1991), 56 | DOI

[3] V. Ya. Prinz, D. Grutzmacher, A. Beyer, C. David, B. Ketterer, E. Deckardt, Nanotechnology, 12 (2001), 1 ; L. A. Chernozatonskii, E. G. Gal'pern, N. R. Serebryanaya, I. V. Stankevich, “Polymers of sungle-wall nanotubes: geometry, diffraction, patterns, and electronic spectrum modeling”, AIP Proc., eds. H. Kuzmany, J. Fink, M. Mehring, S. Roth, AIP, New York, 1999, 284; Н. А. Поклонский, Е. Ф. Кисляков, Г. Г. Федорук, С. А. Вырко, ФТТ, 42:10 (2000), 1911 | DOI

[4] T. Hertel, G. Moos, Phys. Rev. Lett., 84 (2000), 5002 | DOI

[5] V. F. Gantmakher, I. B. Levinson, Rasseyannye nositeli toka v metallakh, Nauka, M., 1984 | MR

[6] H. Jensen, H. Koppe, Ann. Phys., 269 (1971), 77; R. C. T. da Costa, Phys. Rev. A, 23 (1981), 1982 ; 25 (1982), 2893 | DOI | MR | DOI | MR

[7] L. I. Magarill, M. V. Entin, ZhETF, 123:4 (2003), 867; M. V. Entin, L. I. Magarill, Phys. Rev. B, 66 (2002), 205308–1 | DOI

[8] K. Lin, R. L. Jaffe, Phys. Rev. B, {54} (1996), 5720 ; P. C. Schuster, R. L. Jaffe, Ann. Phys., 307 (2003), 132 ; P. Exner, J. Math. Phys. A, 28 (1995), 5323 | DOI | MR | Zbl | DOI | MR | Zbl

[9] B. De Witt, Phys. Rev., 85 (1952), 635

[10] G. Panatti, H. Spohn, S. Teufel, Commun. Math. Phys., 242:3 (2003), 547 ; G. F. Dell'Antonio, L. Tenuta, J. Phys. A, 37 (2004), 5605 ; E-print math-ph/0312034 | DOI | MR | DOI | MR | Zbl

[11] V. P. Maslov, DAN SSSR, 123:4 (1958), 631 | Zbl

[12] E. M. Vorobev, V. P. Maslov, DAN SSSR, 179 (1968), 558 ; V. P. Maslov, Russ. Math. J., 8:1 (2001), 83 | MR | Zbl

[13] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo MGU, M., 1965 | MR

[14] V. P. Maslov, UMN, 38:6 (1983), 3 | MR | Zbl

[15] M. Born, J. R. Oppenheimer, Ann. Phys., 84 (1927), 457 ; М. Борн, Х. Кунь, Динамическая теория кристаллических решеток, ИЛ, М., 1958 | DOI | Zbl

[16] E. M. Lifshits, L. P. Pitaevskii, Statisticheskaya fizika, Ch. 2, Nauka, M., 1978; В. С. Буслаев, ТМФ, 58:2 (1984), 233 ; УМН, 42:6 (1987), 77 ; A. Fedotov, F. Klopp, Commun. Math. Phys., 227:1 (2002), 1 ; В. В. Белов, Изв. вузов. Математика, 6 (1976), 13; J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987 | MR | Zbl | MR | Zbl | DOI | MR | Zbl | Zbl

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

[18] V. G. Vaks, Mezhatomnye vzaimodeistviya i svyaz v tverdykh telakh, IzdAt, M., 2002

[19] S. Yu. Dobrokhotov, DAN SSSR, 269:1 (1983), 76 ; УМН, 39:4 (1984), 125 ; Многофазовые асимптотические решения линейных и нелинейных уравнений в частных производных с малым параметром, Докт. диссертация, Инст. пробл. мех. РАН, М., 1988; Л. В. Берлянд, С. Ю. Доброхотов, ДАН СССР, 296 (1987), 80 ; S. Yu. Dobrokhotov, P. N. Zhevandrov, Russ. J. Math. Phys., 10:1 (2003), 1 | MR | Zbl | MR | Zbl | MR | Zbl

[20] V. V. Belov, S. Yu. Dobrokhotov, S. O. Sinitsyn, Tr. In-ta matematiki i mekhaniki UrO RAN, 9:1 (2003), 15 ; V. V. Belov, S. Yu. Dobrokhotov, S. O. Sinitsyn, T. Ya. Tudorovskiy, Докл. РАН, 393:4 (2003), 460 ; V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskiy, Russ. J. Math. Phys., 11:1 (2004), 109 | MR | Zbl | MR | MR | Zbl

[21] V. P. Maslov, Operatornye metody, Nauka, M., 1973 ; М. В. Карасев, В. П. Маслов, УМН, 39:6 (1984), 115 ; Нелинейные скобки Пуассона. Геометрия и квантование, Наука, М., 1991 | MR | MR | Zbl | MR | Zbl

[22] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[23] L. D. Landau, E. M. Lifshits, Kvantovaya mekhanika. Nerelyativistskaya teoriya, Nauka, M., 1989 ; В. В. Кучеренко, Изв. АН СССР. Сер. матем., 38:3 (1974), 625 ; V. I. Il'ichov, V. A. Rivelis, V. S. Rabinovich, Y. V. Hoha, ДАН СССР, 304:5 (1989), 1123 ; A. Gordon, J. E. Avron, Phys. Rev. Lett., 85:1 (2000), 34 ; J. E. Avron, A. Gordon, Phys. Rev. A, 62 (2000), 062504 ; Y. Colin de Verdier, B. Parisse, Commun. Math. Phys., 205 (1999), 459 | MR | Zbl | DOI | MR | DOI | DOI | MR | Zbl

[24] P. K. Rashevskii, Rimanova geometriya i tenzornyi analiz, Nauka, M., 1967 | MR

[25] T. Hertel, R. E. Walkup, P. Avouris, Phys. Rev. B, 58 (1998), 13870 | DOI

[26] V. A. Geiler, I. Yu. Popov, TMF, 107 (1996), 427 ; В. А. Гейлер, В. А. Маргулис, Физ. техн. полупроводников, 9 (1996), 1141; В. В. Грушин, Матем. заметки, 75:3 (2004), 360 | DOI | MR | Zbl | DOI | MR | Zbl

[27] M. V. Berry, Proc. Roy. Soc. London. A, 392 (1984), 45 | DOI | MR | Zbl