| Title:
|
On the spectrum of Robin Laplacian in a planar waveguide (English) |
| Author:
|
Rossini, Alex Ferreira |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
485-501 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
planar waveguide |
| Keyword:
|
discrete spectrum |
| Keyword:
|
Robin boundary conditions |
| MSC:
|
47B25 |
| MSC:
|
47F05 |
| MSC:
|
49R05 |
| MSC:
|
81Q10 |
| idZBL:
|
Zbl 07088801 |
| idMR:
|
MR3959961 |
| DOI:
|
10.21136/CMJ.2018.0396-17 |
| . |
| Date available:
|
2019-05-24T09:01:14Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147741 |
| . |
| Reference:
|
[1] Adams, R. A.: Sobolev Spaces.Pure and Applied Mathematics 65, Academic Press, New York (1975). Zbl 0314.46030, MR 0450957, 10.1016/S0079-8169(08)61376-8 |
| Reference:
|
[2] Borisov, D., Cardone, G.: Planar waveguide with ``twisted'' boundary conditions: Discrete spectrum.J. Math. Phys. 52 (2011), 123513, 24 pages. Zbl 1273.81100, MR 2907657, 10.1063/1.3670875 |
| Reference:
|
[3] Borisov, D., Krejčiřík, D.: $\mathcal P\mathcal T$-symmetric waveguides.Integral Equations Oper. Theory 62 (2008), 489-515. Zbl 1178.35141, MR 2470121, 10.1007/s00020-008-1634-1 |
| Reference:
|
[4] Chenaud, B., Duclos, P., Freitas, P., Krejčiřík, D.: Geometrically induced discrete spectrum in curved tubes.Differ. Geom. Appl. 23 (2005), 95-105. Zbl 1078.81022, MR 2158038, 10.1016/j.difgeo.2005.05.001 |
| Reference:
|
[5] Oliveira, C. R. de: Intermediate Spectral Theory and Quantum Dynamics.Progress in Mathematical Physics 54, Birkhäuser, Basel (2009). Zbl 1165.47001, MR 2723496, 10.1007/978-3-7643-8795-2 |
| Reference:
|
[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. Zbl 1220.35101, MR 2796223, 10.1016/j.jmaa.2011.03.022 |
| Reference:
|
[7] Dittrich, J., Kříž, J.: Bound states in straight quantum waveguides with combined boundary conditions.J. Math. Phys. 43 (2002), 3892-3915. Zbl 1060.81019, MR 1915632, 10.1063/1.1491597 |
| Reference:
|
[8] Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions.Rev. Math. Phys. 7 (1995), 73-102. Zbl 0837.35037, MR 1310767, 10.1142/S0129055X95000062 |
| Reference:
|
[9] Evans, L. C.: Partial Differential Equations.Graduate Studies in Mathematics 19, AMS, Providence (1998). Zbl 0902.35002, MR 1625845, 10.1090/gsm/019 |
| Reference:
|
[10] Exner, P., Šeba, P.: Bound states in curved quantum waveguides.J. Math. Phys. 30 (1989), 2574-2580. Zbl 0693.46066, MR 1019002, 10.1063/1.528538 |
| Reference:
|
[11] Freitas, P., Krejčiřík, D.: Waveguides with combined Dirichlet and Robin boundary conditions.Math. Phys. Anal. Geom. 9 (2006), 335-352. Zbl 1151.35061, MR 2329432, 10.1007/s11040-007-9015-6 |
| Reference:
|
[12] Goldstone, J., Jaffe, R. L.: Bound states in twisting tubes.Phys. Rev. B. 45 (1992), 14100-14107. 10.1103/PhysRevB.45.14100 |
| Reference:
|
[13] Jílek, M.: Straight quantum waveguide with Robin boundary conditions.SIGMA, Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 108, 12 pages. Zbl 1147.47035, MR 2366914, 10.3842/SIGMA.2007.108 |
| Reference:
|
[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. Zbl 1173.35618, MR 2542572, 10.1051/cocv:2008035 |
| Reference:
|
[15] Krejčiřík, D., Kříž, J.: On the spectrum of curved planar waveguides.Publ. Res. Inst. Math. Sci. 41 (2005), 757-791. Zbl 1113.35143, MR 2154341, 10.2977/prims/1145475229 |
| Reference:
|
[16] Olendski, O., Mikhailovska, L.: Theory of a curved planar waveguide with Robin boundary conditions.Phys. Rev. E. 81 (2010), 036606. 10.1103/PhysRevE.81.036606 |
| Reference:
|
[17] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators.Academic Press, New York (1978). Zbl 0401.47001, MR 0493421 |
| . |
