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Funktsional. Anal. i Prilozhen., 2013, Volume 47, Issue 3, Pages 37–53 (Mi faa3117)  

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

Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: It is shown in the paper that, under several orthogonality and normalization conditions and a proper choice of accessory parameters, a simple eigenvalue lying between thresholds of the continuous spectrum of the Dirichlet problem in a domain with a cylindrical outlet to infinity is not taken out from the spectrum by a small compact perturbation of the Helmholtz operator. The result is obtained by means of an asymptotic analysis of the augmented scattering matrix.

Keywords: continuous spectrum, discrete spectrum, perturbation of eigenvalue, local perturbations of quantum waveguide.


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English version:
Functional Analysis and Its Applications, 2013, 47:3, 195–209

Bibliographic databases:

UDC: 517.984.46+517.958+531.33
Received: 01.06.2011

Citation: S. A. Nazarov, “Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide”, Funktsional. Anal. i Prilozhen., 47:3 (2013), 37–53; Funct. Anal. Appl., 47:3 (2013), 195–209

Citation in format AMSBIB
\by S.~A.~Nazarov
\paper Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide
\jour Funktsional. Anal. i Prilozhen.
\yr 2013
\vol 47
\issue 3
\pages 37--53
\jour Funct. Anal. Appl.
\yr 2013
\vol 47
\issue 3
\pages 195--209

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    This publication is cited in the following articles:
    1. 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
    2. S. A. Nazarov, “The eigenfrequencies of a slightly curved isotropic strip clamped between absolutely rigid profiles”, J. Appl. Math. Mech., 78:4 (2014), 374–383  crossref  mathscinet  isi  elib  scopus
    3. J. T. Kemppainen, S. A. Nazarov, K. M. Ruotsalainen, “Perturbation analysis of embedded eigenvalues for water-waves”, J. Math. Anal. Appl., 427:1 (2015), 399–427  crossref  mathscinet  zmath  scopus
    4. 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
    5. Dhia A.-S.B.-B., Chesnel L., Nazarov S.A., “Non-Scattering Wavenumbers and Far Field Invisibility For a Finite Set of Incident/Scattering Directions”, Inverse Probl., 31:4 (2015), 045006  crossref  mathscinet  zmath  isi  scopus
    6. Chesnel L., Hyvonen N., Staboulis S., “Construction of Indistinguishable Conductivity Perturbations For the Point Electrode Model in Electrical Impedance Tomography”, SIAM J. Appl. Math., 75:5 (2015), 2093–2109  crossref  mathscinet  zmath  isi  elib  scopus
    7. 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
    8. Chesnel L., Nazarov S.A., “Team organization may help swarms of flies to become invisible in closed waveguides”, Inverse Probl. Imaging, 10:4 (2016), 977–1006  crossref  mathscinet  zmath  isi  elib  scopus
    9. Korolkov A.I., Nazarov S.A., Shanin A.V., “Stabilizing solutions at thresholds of the continuous spectrum and anomalous transmission of waves”, ZAMM-Z. Angew. Math. Mech., 96:10 (2016), 1245–1260  crossref  mathscinet  isi  scopus
    10. Nazarov S.A. Ruotsalainen K.M., “A Rigorous Interpretation of Approximate Computations of Embedded Eigenfrequencies of Water Waves”, Z. Anal. ihre. Anwend., 35:2 (2016), 211–242  crossref  mathscinet  zmath  isi  elib  scopus
    11. Kozlov V.A., Nazarov S.A., Orlof A., “Trapped Modes Supported By Localized Potentials in the Zigzag Graphene Ribbon”, C. R. Math., 354:1 (2016), 63–67  crossref  mathscinet  zmath  isi  elib
    12. 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
    13. S. A. Nazarov, “The asymptotic behaviour of the scattering matrix in a neighbourhood of the endpoints of a spectral gap”, Sb. Math., 208:1 (2017), 103–156  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. S. A. Nazarov, “The spectra of rectangular lattices of quantum waveguides”, Izv. Math., 81:1 (2017), 29–90  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. T. Durante, “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. T. Simos, C. Tsitouras, Amer Inst Physics, 2017, UNSP 510003-1  crossref  isi  scopus
    16. V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped modes in zigzag graphene nanoribbons”, Z. Angew. Math. Phys., 68:4 (2017), 78, 31 pp.  crossref  mathscinet  zmath  isi  scopus
    17. F. L. Bakharev, G. Cardone, S. A. Nazarov, J. Taskinen, “Effects of Rayleigh waves on the essential spectrum in perturbed doubly periodic elliptic problems”, Integral Equations Operator Theory, 88:3 (2017), 373–386  crossref  mathscinet  zmath  isi  scopus
    18. F. L. Bakharev, S. G. Matveenko, S. A. Nazarov, “Examples of plentiful discrete spectra in infinite spatial cruciform quantum waveguides”, Z. Anal. Anwend., 36:3 (2017), 329–341  crossref  mathscinet  zmath  isi  scopus
    19. A.-S. Bonnet-Ben Dhia, L. Chesnel, S. A. Nazarov, “Perfect transmission invisibility for waveguides with sound hard walls”, J. Math. Pures Appl., 111 (2018), 79–105  crossref  mathscinet  zmath  isi  scopus
    20. G. Cardone, T. Durante, S. A. Nazarov, “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
    21. V. Ch. Piat, S. A. Nazarov, J. Taskinen, “Embedded eigenvalues for water-waves in a three-dimensional channel with a thin screen”, Q. J. Mech. Appl. Math., 71:2 (2018), 187–220  crossref  mathscinet  isi  scopus
    22. L. Chesnel, S. A. Nazarov, V. Pagneux, “Invisibility and perfect reflectivity in waveguides with finite length branches”, SIAM J. Appl. Math., 78:4 (2018), 2176–2199  crossref  mathscinet  zmath  isi  scopus
    23. 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
    24. S. A. Nazarov, “Enhancement and smoothing of near-threshold wood anomalies in an acoustic waveguide”, Acoust. Phys., 64:5 (2018), 535–547  crossref  isi  scopus
    25. S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations”, J. Math. Sci. (N. Y.), 243:5 (2019), 746–773  mathnet  crossref
    26. 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  elib
    27. L. Chesnel, S. A. Nazarov, “Non reflection and perfect reflection via Fano resonance in waveguides”, Commun. Math. Sci., 16:7 (2018), 1779–1800  crossref  mathscinet  isi  scopus
    28. Chesnel L. Pagneux V., “From Zero Transmission to Trapped Modes in Waveguides”, J. Phys. A-Math. Theor., 52:16 (2019), 165304  crossref  mathscinet  isi  scopus
    29. Sargent C.V., Mestel A.J., “Trapped Modes of the Helmholtz Equation in Infinite Waveguides With Wall Indentations and Circular Obstacles”, IMA J. Appl. Math., 84:2 (2019), 312–344  crossref  isi
    30. S. A. Nazarov, “Zakhvat volny v iskrivlennom tsilindricheskom akusticheskom volnovode s neizmennym secheniem”, Algebra i analiz, 31:5 (2019), 154–183  mathnet
    31. S. A. Nazarov, “Postroenie zakhvachennoi volny na nizkikh chastotakh v uprugom volnovode”, Funkts. analiz i ego pril., 54:1 (2020), 41–57  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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