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TMF, 2005, Volume 145, Number 3, Pages 358–371 (Mi tmf1905)  

This article is cited in 29 scientific papers (total in 29 papers)

Local Perturbations of Quantum Waveguides

R. R. Gadyl'shin

Bashkir State Pedagogical University

Abstract: We give necessary and sufficient conditions for the occurrence of eigenvalues of the Schrodinger operator in strips and cylinders under small localized perturbations. We construct asymptotic approximations for the eigenvalues.

Keywords: Schrodinger operator, waveguide, perturbation, spectrum, asymptotic approximation

DOI: https://doi.org/10.4213/tmf1905

Full text: PDF file (259 kB)
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English version:
Theoretical and Mathematical Physics, 2005, 145:3, 1678–1690

Bibliographic databases:

Received: 16.02.2005

Citation: R. R. Gadyl'shin, “Local Perturbations of Quantum Waveguides”, TMF, 145:3 (2005), 358–371; Theoret. and Math. Phys., 145:3 (2005), 1678–1690

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Grushin, “Asymptotic Behavior of Eigenvalues of the Laplace Operator in Infinite Cylinders Perturbed by Transverse Extensions”, Math. Notes, 81:3 (2007), 291–296  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator in Thin Closed Tubes”, Math. Notes, 83:4 (2008), 463–477  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. V. V. Grushin, “Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes”, Math. Notes, 85:5 (2009), 661–673  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Bikmetov A., Gadyl'shin R., “On quantum waveguide with shrinking potential”, Russ. J. Math. Phys., 17:1 (2010), 19–25  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    5. Gadyl'shin R.R., “On Regular and Singular Perturbations of the Eigenelements of the Laplacian”, Integral Methods in Science and Engineering, Analytic Methods, 2010, 135–148  crossref  mathscinet  zmath  isi
    6. S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide”, Theoret. and Math. Phys., 167:2 (2011), 606–627  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    7. J. H. Videman, V. Chiado' Piat, S. A. Nazarov, “Asymptotics of frequency of a surface wave trapped by a slightly inclined barrier in a liquid layer”, J. Math. Sci. (N. Y.), 185:4 (2012), 536–553  mathnet  crossref  mathscinet
    8. Nazarov S.A., “Trapped Waves in a Cranked Waveguide with Hard Walls”, Acoustical Physics, 57:6 (2011), 764–771  crossref  mathscinet  adsnasa  isi  elib  scopus  scopus
    9. Nazarov S.A., “Asimptotika sobstvennogo chisla zadachi dirikhle v kolenchatom volnovode”, Vestnik Sankt-Peterburgskogo universiteta. Seriya 1: Matematika. Mekhanika. Astronomiya, 2011, no. 3, 29–35  zmath  elib
    10. G. Cardone, S. A. Nazarov, K. Ruotsalainen, “Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide”, Sb. Math., 203:2 (2012), 153–182  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Nazarov S.A., “Asymptotics of Eigenfrequencies in the Spectral Gaps Caused by a Perturbation of a Periodic Waveguide”, Dokl. Math., 86:3 (2012), 871–875  crossref  mathscinet  zmath  isi  elib  scopus
    12. Cardone G., Nazarov S.A., Ruotsalainen K., “Bound States of a Converging Quantum Waveguide”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 47:1 (2013), 305–315  crossref  mathscinet  zmath  isi  scopus  scopus
    13. S. A. Nazarov, “Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide”, Funct. Anal. Appl., 47:3 (2013), 195–209  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. Nazarov S.A., “Gaps and Eigenfrequencies in the Spectrum of a Periodic Acoustic Waveguide”, Acoust. Phys., 59:3 (2013), 272–280  crossref  adsnasa  isi  elib  scopus  scopus
    15. S. A. Nazarov, “Bounded solutions in a $\mathrm{T}$-shaped waveguide and the spectral properties of the Dirichlet ladder”, Comput. Math. Math. Phys., 54:8 (2014), 1261–1279  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. Nazarov S.A., Ruotsalainen K., Uusitalo P., “Bound States of Waveguides With Two Right-Angled Bends”, J. Math. Phys., 56:2 (2015), 021505  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    17. S. A. Nazarov, “Scattering anomalies in a resonator above the thresholds of the continuous spectrum”, Sb. Math., 206:6 (2015), 782–813  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Nazarov S.A., “Near-threshold effects of the scattering of waves in a distorted elastic two-dimensional waveguide”, Pmm-J. Appl. Math. Mech., 79:4 (2015), 374–387  crossref  mathscinet  isi  scopus
    19. S. A. Nazarov, “Almost standing waves in a periodic waveguide with a resonator and near-threshold eigenvalues”, St. Petersburg Math. J., 28:3 (2017), 377–410  mathnet  crossref  mathscinet  isi  elib
    20. Nazarov S.A., Ruotsalainen K.M., Silvola M., “Trapped Modes in Piezoelectric and Elastic Waveguides”, J. Elast., 124:2 (2016), 193–223  crossref  mathscinet  zmath  isi  elib  scopus
    21. 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
    22. Lampart J., Teufel S., “The adiabatic limit of Schrödinger operators on fibre bundles”, Math. Ann., 367:3-4 (2017), 1647–1683  crossref  mathscinet  zmath  isi  scopus
    23. I. Kh. Khusnullin, “Vozmuschenie volnovoda uzkim potentsialom”, Tr. IMM UrO RAN, 23, no. 2, 2017, 274–284  mathnet  crossref  elib
    24. Durante T., “Waveguides With a Box-Shaped Perturbation: Eigenvalues of the Neumann Problem”, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2016 (ICNAAM-2016), AIP Conference Proceedings, 1863, eds. Simos T., Tsitouras C., Amer Inst Physics, 2017, UNSP 510003-1  crossref  isi  scopus  scopus
    25. Cardone G., Durante T., Nazarov S.A., “Embedded Eigenvalues of the Neumann Problem in a Strip With a Box-Shaped Perturbation”, J. Math. Pures Appl., 112 (2018), 1–40  crossref  mathscinet  zmath  isi  scopus  scopus
    26. 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  scopus  scopus
    27. 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
    28. 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
    29. S. A. Nazarov, “Various manifestations of Wood anomalies in locally distorted quantum waveguides”, Comput. Math. Math. Phys., 58:11 (2018), 1838–1855  mathnet  crossref  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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