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Mat. Sb., 2006, Volume 197, Number 4, Pages 3–32 (Mi msb1545)  

This article is cited in 31 scientific papers (total in 31 papers)

Discrete spectrum of an asymmetric pair of waveguides coupled through a window

D. I. Borisovab

a Bashkir State Pedagogical University
b Nuclear Physics Institute, Academy of Sciences of the Czech Republic

Abstract: 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.

DOI: https://doi.org/10.4213/sm1545

Full text: PDF file (655 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2006, 197:4, 475–504

Bibliographic databases:

UDC: 517.958
MSC: 47F05, 47N50, 35P20
Received: 17.08.2004 and 24.11.2005

Citation: D. I. Borisov, “Discrete spectrum of an asymmetric pair of waveguides coupled through a window”, Mat. Sb., 197:4 (2006), 3–32; Sb. Math., 197:4 (2006), 475–504

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Borisov D., Exner P., “Distant perturbation asymptotics in window-coupled waveguides. I. The nonthreshold case”, J. Math. Phys. (N. Y.), 47:11 (2006), 113502, 24 pp.  crossref  mathscinet  zmath  adsnasa  isi
    2. Borisov D., “The spectrum of two quantum layers coupled by a window”, J. Phys. A, 40:19 (2007), 5045–5066  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Kondej S., Veselić I., “Spectral gap of segments of periodic waveguides”, Lett. Math. Phys., 79:1 (2007), 95–98  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Krejčiřik D., Tater M., “Non-Hermitian spectral effects in a PT-symmetric waveguide”, J. Phys. A, 41:24 (2008), 244013, 14 pp.  crossref  mathscinet  zmath  adsnasa  isi
    5. Borisov D., Krejčiřik D., “$\mathscr{PT}$-symmetric waveguides”, Integral Equations Operator Theory, 62:4 (2008), 489–515  crossref  mathscinet  zmath  isi  elib
    6. Borisov D., Cardone G., “Homogenization of the planar waveguide with frequently alternating boundary conditions”, J. Phys. A, 42:36 (2009), 365205, 21 pp.  crossref  mathscinet  zmath  isi
    7. Borisov B., Bunoiu R., Cardone G., “On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition”, Ann. Henri Poincaré, 11:8 (2010), 1591–1627  crossref  mathscinet  zmath  adsnasa  isi
    8. Borisov D., Veselić I., “Low lying spectrum of weak-disorder quantum waveguides”, J. Stat. Phys., 142:1 (2011), 58–77  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. D. I. Borisov, “On the spectrum of a two-dimensional periodic operator with a small localized perturbation”, Izv. Math., 75:3 (2011), 471–505  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. S. A. Nazarov, “Asymptotics of trapped modes and eigenvalues below the continuous spectrum of a waveguide with a thin shielding obstacle”, St. Petersburg Math. J., 23:3 (2012), 571–601  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    11. Borisov D., Cardone G., “Planar waveguide with “twisted” boundary conditions: Discrete spectrum”, J. Math. Phys., 52:12 (2011), 123513, 24 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Borisov D., Cardone G., “Planar waveguide with “twisted” boundary conditions: Small width”, J. Math. Phys., 53:2 (2012), 023503, 22 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. D. I. Borisov, “On a $\mathcal{PT}$-symmetric waveguide with a pair of small holes”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 5–21  mathnet  crossref  isi  elib
    14. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics”, Z. Angew. Math. Phys, 2012  crossref  mathscinet  isi
    15. D. Borisov, P. Exner, A. Golovina, “Tunneling resonances in systems without a classical trapping”, J. Math. Phys, 54:1 (2013), 012102  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. S. A. Nazarov, “Asymptotics of an Eigenvalue on the Continuous Spectrum of Two Quantum Waveguides Coupled through Narrow Windows”, Math. Notes, 93:2 (2013), 266–281  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    17. A. M. Golovina, “On the spectrum of elliptic operators with distant perturbation in the space”, St. Petersburg Math. J., 25:5 (2014), 735–754  mathnet  crossref  mathscinet  zmath  isi  elib
    18. Borisov D., Veselic I., “Low Lying Eigenvalues of Randomly Curved Quantum Waveguides”, J. Funct. Anal., 265:11 (2013), 2877–2909  crossref  mathscinet  zmath  isi  elib
    19. D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide”, Ufa Math. J., 6:1 (2014), 29–55  mathnet  crossref  isi  elib
    20. T. F. Sharapov, “On the resolvent of multidimensional operators with frequently changing boundary conditions in the case of the homogenized Dirichlet condition”, Sb. Math., 205:10 (2014), 1492–1527  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. D. I. Borisov, “Perturbation of Threshold of Essential Spectrum for Waveguides with Windows. I: Decreasing Resonance Solutions”, J Math Sci, 205:2 (2015), 141  crossref  mathscinet  zmath  elib
    22. D.I. Borisov, R. Kh. Karimov, T. F. Sharapov, “Initial length scale estimate for waveguides with some random singular potentials”, Ufa Math. J., 7:2 (2015), 33–54  mathnet  crossref  isi  elib
    23. D.I. Borisov, “The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883  mathnet  crossref  crossref  mathscinet  isi  elib
    24. Bikmetov A.R., Gadyl'shin R.R., “On local perturbations of waveguides”, Russ. J. Math. Phys., 23:1 (2016), 1–18  crossref  mathscinet  zmath  isi  scopus
    25. Borisov D.I., “Creation of spectral bands for a periodic domain with small windows”, Russ. J. Math. Phys., 23:1 (2016), 19–34  crossref  mathscinet  zmath  isi  scopus
    26. Borisov D., Cardone G., Durante T., “Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve”, Proc. R. Soc. Edinb. Sect. A-Math., 146:6 (2016), 1115–1158  crossref  mathscinet  zmath  isi  scopus
    27. Borisov D., Golovina A., Veselic I., “Quantum Hamiltonians with Weak Random Abstract Perturbation. I. Initial Length Scale Estimate”, Ann. Henri Poincare, 17:9 (2016), 2341–2377  crossref  mathscinet  zmath  isi  scopus
    28. Borisov D.I., Dmitriev S.V., “On the Spectral Stability of Kinks in 2D Klein-Gordon Model with Parity-Time-Symmetric Perturbation”, Stud. Appl. Math., 138:3 (2017), 317–342  crossref  mathscinet  zmath  isi  scopus
    29. Piat V.Ch., Nazarov S.A., Taskinen J., “Embedded Eigenvalues Forwater-Waves in Athree-Dimensional Channel With Athin Screen”, Q. J. Mech. Appl. Math., 71:2 (2018), 187–220  crossref  mathscinet  isi
    30. S. A. Nazarov, “Transmission of waves through a small aperture in the cross-wall in an acoustic waveguide”, Siberian Math. J., 59:1 (2018), 85–101  mathnet  crossref  crossref  isi  elib
    31. S. A. Nazarov, “Asimptotika sobstvennykh chisel vnutri lakun spektra periodicheskikh volnovodov s malymi singulyarnymi vozmuscheniyami”, Matematicheskie voprosy teorii rasprostraneniya voln. 48, Posvyaschaetsya pamyati Aleksandra Pavlovicha KAChALOVA, Zap. nauchn. sem. POMI, 471, POMI, SPb., 2018, 168–210  mathnet
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