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Keywords:
planar waveguide; discrete spectrum; Robin boundary conditions
Summary:
We consider the Laplace operator in a planar waveguide, i.e. an infinite two-dimensional straight strip of constant width, with Robin boundary conditions. We study the essential spectrum of the corresponding Laplacian when the boundary coupling function has a limit at infinity. Furthermore, we derive sufficient conditions for the existence of discrete spectrum.
References:
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65, Academic Press, New York (1975). DOI 10.1016/S0079-8169(08)61376-8 | MR 0450957 | Zbl 0314.46030
[2] Borisov, D., Cardone, G.: Planar waveguide with ``twisted'' boundary conditions: Discrete spectrum. J. Math. Phys. 52 (2011), 123513, 24 pages. DOI 10.1063/1.3670875 | MR 2907657 | Zbl 1273.81100
[3] Borisov, D., Krejčiřík, D.: $\mathcal P\mathcal T$-symmetric waveguides. Integral Equations Oper. Theory 62 (2008), 489-515. DOI 10.1007/s00020-008-1634-1 | MR 2470121 | Zbl 1178.35141
[4] Chenaud, B., Duclos, P., Freitas, P., Krejčiřík, D.: Geometrically induced discrete spectrum in curved tubes. Differ. Geom. Appl. 23 (2005), 95-105. DOI 10.1016/j.difgeo.2005.05.001 | MR 2158038 | Zbl 1078.81022
[5] Oliveira, C. R. de: Intermediate Spectral Theory and Quantum Dynamics. Progress in Mathematical Physics 54, Birkhäuser, Basel (2009). DOI 10.1007/978-3-7643-8795-2 | MR 2723496 | Zbl 1165.47001
[6] Oliveira, C. R. de, Verri, A. A.: On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes. J. Math. Anal. Appl. 381 (2011), 454-468. DOI 10.1016/j.jmaa.2011.03.022 | MR 2796223 | Zbl 1220.35101
[7] Dittrich, J., Kříž, J.: Bound states in straight quantum waveguides with combined boundary conditions. J. Math. Phys. 43 (2002), 3892-3915. DOI 10.1063/1.1491597 | MR 1915632 | Zbl 1060.81019
[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 10.1142/S0129055X95000062 | MR 1310767 | Zbl 0837.35037
[9] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19, AMS, Providence (1998). DOI 10.1090/gsm/019 | MR 1625845 | Zbl 0902.35002
[10] Exner, P., Šeba, P.: Bound states in curved quantum waveguides. J. Math. Phys. 30 (1989), 2574-2580. DOI 10.1063/1.528538 | MR 1019002 | Zbl 0693.46066
[11] Freitas, P., Krejčiřík, D.: Waveguides with combined Dirichlet and Robin boundary conditions. Math. Phys. Anal. Geom. 9 (2006), 335-352. DOI 10.1007/s11040-007-9015-6 | MR 2329432 | Zbl 1151.35061
[12] Goldstone, J., Jaffe, R. L.: Bound states in twisting tubes. Phys. Rev. B. 45 (1992), 14100-14107. DOI 10.1103/PhysRevB.45.14100
[13] Jílek, M.: Straight quantum waveguide with Robin boundary conditions. SIGMA, Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 108, 12 pages. DOI 10.3842/SIGMA.2007.108 | MR 2366914 | Zbl 1147.47035
[14] Krejčiřík, D.: Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM, Control Optim. Calc. Var. 15 (2009), 555-568. DOI 10.1051/cocv:2008035 | MR 2542572 | Zbl 1173.35618
[15] Krejčiřík, D., Kříž, J.: On the spectrum of curved planar waveguides. Publ. Res. Inst. Math. Sci. 41 (2005), 757-791. DOI 10.2977/prims/1145475229 | MR 2154341 | Zbl 1113.35143
[16] Olendski, O., Mikhailovska, L.: Theory of a curved planar waveguide with Robin boundary conditions. Phys. Rev. E. 81 (2010), 036606. DOI 10.1103/PhysRevE.81.036606
[17] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York (1978). MR 0493421 | Zbl 0401.47001
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