Geodesic
Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder
Matematičeskie zametki, Tome 77 (2005) no. 5, pp. 656-664 
V. V. Grushin. Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder. Matematičeskie zametki, Tome 77 (2005) no. 5, pp. 656-664. http://geodesic.mathdoc.fr/item/MZM_2005_77_5_a1/
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     title = {Asymptotic {Behavior} of the {Eigenvalues} of the {Schr\"odinger} {Operator} with {Transversal} {Potential} in a {Weakly} {Curved} {Infinite} {Cylinder}},
     journal = {Matemati\v{c}eskie zametki},
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In this paper, we derive sufficient conditions for the existence of an eigenvalue for the Laplace and the Schrödinger operators with transversal potential for homogeneous Dirichlet boundary conditions in a tube, i.e., in a curved and twisted infinite cylinder. For tubes with small curvature and small internal torsion, we derive an asymptotic formula for the eigenvalue of the problem. We show that, under certain relations between the curvature and the internal torsion of the tube, the above operators possess no discrete spectrum.

[1] Maslov V. P., “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 | Zbl

[2] Maslov V. P., “Mathematical Aspects of Integral Optics”, Russian J. Math. Phys., 8 (2001), 83–180 | MR

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

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

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

[6] Belov V. V., Dobrokhotov S. Yu., Tudorovskii T. Ya., “Quantum and classical dynamics of electron in thin curved tubes with spin and external electromagnetic fields taken into account”, Russian J. Math. Phys., 11:1 (2004), 109–118 | MR

[7] Belov V. V., Dobrokhotov S. Yu., Tudorovskii T. Ya., “Asimptoticheskie resheniya nerelyativistskikh uravnenii kvantovoi mekhaniki v iskrivlennykh nanotrubkakh, 1”, TMF, 141:2 (2004), 267–303 | MR

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

[9] Exner P., “Bound States in curved quantum waveguides”, J. Math. Phys., 30 (1989), 2574–2580 | DOI

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

[11] Exner P., “A quantum pipette”, J. Math. Phys. A, 28 (1995), 5323–5330 | DOI

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

[13] Entin M. V., Magarill L. I., “Electrons in twisted quantum wire”, Phys. Rev. B, 66:1–5 (2002), 205308-1–205308-5 | DOI

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

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

[16] Khërmander L., Lineinye differentsialnye operatory, Mir, M., 1965

[17] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi i uglovymi tochkami”, Tr. MMO, 16, URSS, M., 1967, 207–318

[18] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. T. 4. Analiz operatorov, Mir, M., 1982

[19] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. T. 1. Funktsionalnyi analiz, Mir, M., 1982

[20] Gokhberg I. Ts., “O lineinykh operatorakh, analiticheski zavisyaschikh ot parametra”, Dokl. AN SSSR, 78:4 (1951), 629–632

[21] Blekher P. M., “Ob operatorakh, meromorfno zavisyaschikh ot parametra”, Vestn. MGU. Ser. 1. Matem., mekh., 1965, no. 5, 30–36