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TMF, 2011, Volume 167, Number 2, Pages 239–263 (Mi tmf6637)  

This article is cited in 43 scientific papers (total in 43 papers)

Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide

S. A. Nazarov

Institute of Mechanical Engineering Problems, RAS, St.~Petersburg, Russia

Abstract: We establish that by choosing a smooth local perturbation of the boundary of a planar quantum waveguide, we can create an eigenvalue near any given threshold of the continuous spectrum and the corresponding trapped wave exponentially decaying at infinity. Based on an analysis of an auxiliary object, a unitary augmented scattering matrix, we asymptotically interpret Wood's anomalies, the phenomenon of fast variations in the diffraction pattern due to variations in the near-threshold wave frequency.

Keywords: quantum waveguide, regular perturbation of the boundary, asymptotic expansion of eigenvalue on the continuous spectrum, Wood's anomalies


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English version:
Theoretical and Mathematical Physics, 2011, 167:2, 606–627

Bibliographic databases:

Document Type: Article
Received: 12.11.2010

Citation: S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide”, TMF, 167:2 (2011), 239–263; Theoret. and Math. Phys., 167:2 (2011), 606–627

Citation in format AMSBIB
\by S.~A.~Nazarov
\paper Asymptotic expansions of eigenvalues in the~continuous spectrum of a~regularly perturbed quantum waveguide
\jour TMF
\yr 2011
\vol 167
\issue 2
\pages 239--263
\jour Theoret. and Math. Phys.
\yr 2011
\vol 167
\issue 2
\pages 606--627

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    This publication is cited in the following articles:
    1. Aya H. Cano R. Zhevandrov P., “Scattering and embedded trapped modes for an infinite nonhomogeneous Timoshenko beam”, J. Engrg. Math., 77:1 (2012), 87–104  crossref  mathscinet  zmath  isi  scopus
    2. Nazarov S.A., “Asymptotics of the reflection coefficient at critical frequencies in a narrowing waveguide”, Russ. J. Math. Phys., 19:2 (2012), 216–233  crossref  mathscinet  zmath  isi  elib  scopus
    3. 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
    4. S. A. Nazarov, “Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum”, Comput. Math. Math. Phys., 53:6 (2013), 702–720  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. S. A. Nazarov, “Nonreflecting distortions of an isotropic strip clamped between rigid punches”, Comput. Math. Math. Phys., 53:10 (2013), 1512–1522  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. Dhia A. -S. Bonnet-Ben, Nazarov S.A., “Obstacles in Acoustic Waveguides Becoming “Invisible” at Given Frequencies”, Acoust. Phys., 59:6 (2013), 633–639  crossref  adsnasa  isi  scopus
    7. S. A. Nazarov, “Asymptotics of eigenvalues of the Dirichlet problem in a skewed $\mathcal{T}$-shaped waveguide”, Comput. Math. Math. Phys., 54:5 (2014), 793–814  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. S. A. Nazarov, “Umov-Mandelshtam radiation conditions in elastic periodic waveguides”, Sb. Math., 205:7 (2014), 953–982  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. 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
    10. Nazarov S.A., “The Eigenfrequencies of a Slightly Curved Isotropic Strip Clamped Between Absolutely Rigid Profiles”, Pmm-J. Appl. Math. Mech., 78:4 (2014), 374–383  crossref  mathscinet  isi  scopus
    11. Romero Rodriguez M.I., Zhevandrov P., “Trapped Modes and Scattering For Oblique Waves in a Two-Layer Fluid”, J. Fluid Mech., 753 (2014), 427–447  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. Nazarov S.A., “Nonreflection and Trapping of Elastic Waves in a Slightly Curved Isotropic Strip”, Dokl. Phys., 59:3 (2014), 139–143  crossref  mathscinet  adsnasa  isi  scopus
    13. 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
    14. Bonnet-Ben Dhia A.-S., Nazarov S.A., Taskinen J., “Underwater Topography Invisible For Surface Waves At Given Frequencies”, Wave Motion, 57 (2015), 129–142  crossref  mathscinet  isi  scopus
    15. Bakharev F.L., Matveenko S.G., Nazarov S.A., “Spectra of Three-Dimensional Cruciform and Lattice Quantum Waveguides”, Dokl. Math., 92:1 (2015), 514–518  crossref  mathscinet  zmath  isi  elib  scopus
    16. Kemppainen J.T., Nazarov S.A., Ruotsalainen K.M., “Perturbation Analysis of Embedded Eigenvalues For Water-Waves”, J. Math. Anal. Appl., 427:1 (2015), 399–427  crossref  mathscinet  zmath  isi  elib  scopus
    17. 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
    18. 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
    19. S. A. Nazarov, “Discrete spectrum of cranked quantum and elastic waveguides”, Comput. Math. Math. Phys., 56:5 (2016), 864–880  mathnet  crossref  crossref  isi  elib
    20. 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
    21. 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
    22. 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
    23. 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
    24. 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
    25. 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
    26. Nazarov S.A., Ruotsalainen K., Uusitalo P., “Multifarious Transmission Conditions in the Graph Models of Carbon Nano-Structures”, Mater. Phys. Mech., 29:2 (2016), 107–115  isi
    27. Nazarov S.A., Ruotsalainen K., Uusitalo P., “Localized Waves in Carbon Nano-Structures With Connected and Disconnected Open Waveguides”, Mater. Phys. Mech., 29:2 (2016), 116–124  isi
    28. 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
    29. 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
    30. S. A. Nazarov, “Open waveguides in a thin Dirichlet lattice: II. Localized waves and radiation conditions”, Comput. Math. Math. Phys., 57:2 (2017), 236–252  mathnet  crossref  crossref  isi  elib
    31. F. L. Bakharev, S. G. Matveenko, S. A. Nazarov, “Rectangular lattices of cylindrical quantum waveguides. I. Spectral problems in a finite cross”, St. Petersburg Math. J., 29:3 (2018), 423–437  mathnet  crossref  mathscinet  isi  elib
    32. Nazarov S.A., “Wave Scattering in the Joint of a Straight and a Periodic Waveguide”, Pmm-J. Appl. Math. Mech., 81:2 (2017), 129–147  crossref  mathscinet  isi  scopus
    33. Kozlov V.A., Nazarov S.A., Orlof A., “Trapped Modes in Zigzag Graphene Nanoribbons”, Z. Angew. Math. Phys., 68:4 (2017), 78  crossref  mathscinet  zmath  isi  scopus
    34. Bakharev F.L., Cardone G., Nazarov S.A., Taskinen J., “Effects of Rayleigh Waves on the Essential Spectrum in Perturbed Doubly Periodic Elliptic Problems”, Integr. Equ. Oper. Theory, 88:3 (2017), 373–386  crossref  mathscinet  zmath  isi  scopus
    35. Bakharev F.L., Matveenko S.G., Nazarov S.A., “Examples of Plentiful Discrete Spectra in Infinite Spatial Cruciform Quantum Waveguides”, Z. Anal. ihre. Anwend., 36:3 (2017), 329–341  crossref  mathscinet  zmath  isi  scopus
    36. Bonnet-Ben Dhia A.-S., Chesnel L., Nazarov S.A., “Perfect Transmission Invisibility For Waveguides With Sound Hard Walls”, J. Math. Pures Appl., 111 (2018), 79–105  crossref  mathscinet  zmath  isi  scopus
    37. 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
    38. 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
    39. Chesnel L., Pagneux V., “Simple Examples of Perfectly Invisible and Trapped Modes in Waveguides”, Q. J. Mech. Appl. Math., 71:3 (2018), 297–315  crossref  mathscinet  isi
    40. Chesnel L., Nazarov S.A., Pagneux V., “Invisibility and Perfect Reflectivity in Waveguides With Finite Length Branches”, SIAM J. Appl. Math., 78:4 (2018), 2176–2199  crossref  mathscinet  zmath  isi  scopus
    41. 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
    42. Nazarov S.A., “Enhancement and Smoothing of Near-Threshold Wood Anomalies in An Acoustic Waveguide”, Acoust. Phys., 64:5 (2018), 535–547  crossref  isi  scopus
    43. 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
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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