Geodesic
Discrete spectrum of an asymmetric pair of waveguides coupled through a window
Sbornik. Mathematics, Tome 197 (2006) no. 4, pp. 475-504 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper one analyses the discrete spectrum of an asymmetric pair of two-dimensional quantum waveguides with common boundary in which a window of finite size is made. The phenomenon of new eigenvalues arising at the boundary of the essential spectrum as the length of the window passes over critical values is considered. For the newly arising eigenvalues one constructs asymptotic expansions with respect to the small parameter equal to the difference between the window length and the closest critical value. The behaviour of the spectrum under an unrestricted growth of the length of the window is also under investigation; asymptotic expansions for eigenvalues with respect to the large parameter, the length of the window, are constructed. Bibliography: 22 titles. 
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     title = {Discrete spectrum of an asymmetric pair of waveguides coupled through a~window},
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D. I. Borisov. Discrete spectrum of an asymmetric pair of waveguides coupled through a window. Sbornik. Mathematics, Tome 197 (2006) no. 4, pp. 475-504. http://geodesic.mathdoc.fr/item/SM_2006_197_4_a0/

[1] P. Exner, P. Šeba, M. Tater, D. Vaněk, “Bound states and scattering in quantum waveguides coupled laterally through a boundary window”, J. Math. Phys., 37:10 (1996), 4867–4887 | DOI | MR | Zbl

[2] Y. Hirayama, Y. Tokura, A. D. Wieck, S. Koch, R. J. Haug, K. von Klitzing, K. Ploog, “Transport characteristics of a window-coupled in-plane-gated wire system”, Phys. Rev. B, 48:11 (1993), 7991–7998 | DOI

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

[4] P. Exner, S. Vugalter, “Asymptotics estimates for bound states in quantum waveguides coupled laterally through a narrow window”, Ann. Inst. H. Poincaré Phys. Théor., 65:1 (1996), 109–123 | MR | Zbl

[5] P. Exner, S. Vugalter, “Bound-state asymptotic estimate for window-coupled Dirichlet strips and layers”, J. Phys. A, 30:22 (1997), 7863–7878 | DOI | MR | Zbl

[6] I. Yu. Popov, “Asymptotics of bound states for laterally coupled waveguides”, Rep. Math. Phys., 43:3 (1999), 427–437 | DOI | MR | Zbl

[7] R. Gadyl'shin, “On regular and singular perturbation of acoustic and quantum waveguides”, C. R. Mecanique, 332:8 (2004), 647–652 | DOI

[8] D. Borisov, P. Exner, R. Gadyl'shin, “Geometric coupling thresholds in a two-dimensional strip”, J. Math. Phys., 43:12 (2002), 6265–6278 | DOI | MR | Zbl

[9] Ch. Kunze, “Leaky and mutually coupled wires”, Phys. Rev. B, 48:19 (1993), 14338–14346 | DOI

[10] J. Dittrich, J. Kříž, “Bound states in straight quantum waveguide with combined boundary condition”, J. Math. Phys., 43:8 (2002), 3892–3915 | DOI | MR | Zbl

[11] D. Borisov, T. Ekholm, H. Kovařík, “Spectrum of the magnetic Schrödinger operator in a waveguide with combined boundary conditions”, Ann. Henri Poincaré, 6:2 (2005), 327–342 | DOI | MR | Zbl

[12] M. Sh. Birman, “Vozmuscheniya nepreryvnogo spektra singulyarnogo ellipticheskogo operatora pri vozmuschenii granitsy i granichnykh uslovii”, Vestn. LGU, 17:1 (1962), 22–55 | MR | Zbl

[13] R. A. Adams, Sobolev spaces, Pure Appl. Math., 65, Academic Press, New York, 1975 | MR | Zbl

[14] V. P. Mikhailov, Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, Fizmatlit, M., 1976 | MR | Zbl

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

[16] J. Blank, P. Exner, M. Havliček, Hilbert space operators in quantum physics, AIP Press, New York, 1994 | MR | Zbl

[17] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | Zbl

[18] D. Borisov, P. Exner, “Exponential splitting of bound states in a waveguide with a pair of distant windows”, J. Phys. A, 37:10 (2004), 3411–3428 | DOI | MR | Zbl

[19] E. Sanches-Palensiya, Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[20] R. R. Gadylshin, “O lokalnykh vozmuscheniyakh operatora Shrëdingera na osi”, TMF, 132:1 (2002), 97–104 | MR | Zbl

[21] R. R. Gadylshin, “O lokalnykh vozmuscheniyakh operatora Shrëdingera na ploskosti”, TMF, 138:1 (2004), 41–54 | MR

[22] A. M. Ilin, Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, Fizmatlit, M., 1989 | MR