Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes

V. V. Grushin

Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 687-701.

Résumé

In this paper, we obtain an asymptotic expansion for the eigenvalues of the Laplace operator with zero Dirichlet conditions in tubes, i.e., in infinite bent cylinders with internal torsion under uniform contraction of their cross-sections, with respect to a small parameter characterizing the transverse dimensions of the tube. A method of reducing the problem of determining the eigenvalues to the solution of an implicit equation is proposed.

Détail

Keywords: eigenvalues of the Laplace operator, Dirichlet condition, thin infinite tube, Schrödinger operator, Maslov canonical operator, quantum waveguide.
Mots-clés : Frenet equations

Bibliographie

[1] V. P. Maslov, “Asimptotika sobstvennykh funktsii uravneniya $\Delta u+k^2u=0$ s kraevymi usloviyami na ekvidistantnykh krivykh i rasseyanie elektromagnitnykh voln v volnovode”, Dokl. AN SSSR, 123:4 (1958), 631–633 | MR | Zbl

[2] V. P. Maslov, “Mathematical aspects of integral optics”, Russ. J. Math. Phys., 8:1 (2001), 83–105 | MR | Zbl

[3] V. P. Maslov, E. M. Vorobev, “Ob odnomodovykh otkrytykh rezonatorakh”, Dokl. AN SSSR, 179:3 (1968), 558–561

[4] V. V. Belov, S. Yu. Dobrokhotov, S. O. Sinitsyn, “Asimptoticheskie resheniya uravneniya Shrëdingera v tonkikh trubkakh”, Tr. In-ta matem. i mekh. UrO RAN, 9:1 (2003), 15–25 | MR | Zbl

[5] V. V. Belov, S. Yu. Dobrokhotov, S. O. Sinitsyn, T. Ya. Tudorovskii, “Kvaziklassicheskoe priblizhenie i kanonicheskii operator Maslova dlya nerelyativistskikh uravnenii kvantovoi mekhaniki v nanotrubkakh”, Dokl. RAN, 393:4 (2003), 460–464 | MR

[6] V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Quantum and classical dynamics of electron in thin curved tubes with spin and external electromagnetic fields taken into account”, Russ. J. Math. Phys., 11:1 (2004), 109–119 | MR | Zbl

[7] V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Asimptoticheskie resheniya nerelyativistskikh uravnenii kvantovoi mekhaniki v iskrivlennykh nanotrubkakh. I. Reduktsiya k prostranstvenno-odnomernym uravneniyam”, TMF, 141:2 (2004), 267–303 | MR

[8] V. V. Belov, S. Yu. Dobrokhotov, V. P. Maslov, T. Ya. Tudorovskii, “Obobschennyi adiabaticheskii printsip dlya opisaniya dinamiki elektrona v iskrivlennykh nanostrukturakh”, UFN, 175:9 (2005), 1004–1010 | DOI

[9] P. Duclos, P. Exner, “Curvature-induced bound states in quantum waveguides in two and three dimensions”, Rev. Math. Phys., 7:1 (1995), 73–102 | DOI | MR | Zbl

[10] P. Exner, P. Šeba, “Bound states in curved quantum waveguides”, J. Math. Phys., 30:11 (1989), 2574–2580 | DOI | MR | Zbl

[11] P. Exner, “Bound states in quantum waveguides of a slowly decaying curvature”, J. Math. Phys., 34:1 (1993), 23–28 | DOI | MR | Zbl

[12] P. Exner, “A quantum pipette”, J. Phys. A, 28:18 (1995), 5323–5330 | DOI | MR | Zbl

[13] W. Bulla, F. Gesztesy, W. Renger, B. Simon, “Weakly couped bound states in quantum waveguides”, Proc. Amer. Math. Soc., 125:5 (1997), 1487–1495 | DOI | MR | Zbl

[14] P. Exner, S. A. Vugalter, “Bound states in a locally deformed waveguide: the critical case”, Lett. Math. Phys., 39:1 (1997), 59–68 | DOI | MR | Zbl

[15] D. Borisov, P. Exner, R. Gadyl'shin, D. Krejčiřík, “Bound states in a weakly deformed strips and layers”, Ann. Henri Poincaré, 2:3 (2001), 553–572 | DOI | MR | Zbl

[16] V. V. Grushin, “O sobstvennykh znacheniyakh finitno vozmuschennogo operatora Laplasa v beskonechnykh tsilindricheskikh oblastyakh”, Matem. zametki, 75:3 (2004), 360–371 | MR | Zbl

[17] V. V. Grushin, “Asimptoticheskoe povedenie sobstvennykh znachenii operatora Shrëdingera s poperechnym potentsialom v slabo iskrivlennykh beskonechnykh tsilindrakh”, Matem. zametki, 77:5 (2005), 656–664 | MR | Zbl

[18] R. R. Gadylshin, “O lokalnykh vozmuscheniyakh kvantovykh volnovodov”, TMF, 145:3 (2005), 358–371 | MR

[19] L. I. Magarill, M. V. Entin, “Elektrony v krivolineinoi kvantovoi provoloke”, ZhETF, 123:4 (2003), 867–876

[20] L. Hörmander, Linear Partial Differential Operators, Grundlehren Math. Wiss., 116, Springer-Verlag, Berlin, 1963 ; Л. Хермандер, Линейные дифференциальные операторы с частными производными, Мир, М., 1965 | MR | Zbl | MR | Zbl

[21] M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978. ; М. Рид, Б. Саймон, Методы современной математической физики. Т. 4. Анализ операторов, Мир, М., 1982 | MR | Zbl | MR | Zbl

[22] B. Chenaud, P. Duclos, P. Freitas, D. Krejčiřík, “Geometrically induced discrete spectrum in curved tubes”, Differential Geom. Appl., 23:2 (2005), 95–105 | DOI | MR | Zbl

[23] M. S. Agranovich, M. I. Vishik, “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, UMN, 19:3 (1964), 53–161 | MR | Zbl

[24] M. Reed, B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1972 ; М. Рид, Б. Саймон, Методы современной математической физики. Т. 1. Функциональный анализ, Мир, М., 1977 | MR | Zbl | MR

[25] V. V. Grushin, “Ob odnom klasse psevdodifferentsialnykh operatorov, vyrozhdayuschikhsya na podmnogoobrazii”, Matem. sb., 84:2 (1971), 163–195 | MR | Zbl