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Mat. Zametki, 2004, Volume 75, Issue 3, Pages 360–371 (Mi mz40)  

This article is cited in 28 scientific papers (total in 28 papers)

On the Eigenvalues of Finitely Perturbed Laplace Operators in Infinite Cylindrical Domains

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: In this paper, sufficient conditions for the existence of eigenvalues of a finitely perturbed Laplace operator in an infinite cylindrical domain and their asymptotics in the small parameter are given. Similar questions for tubes, i.e., deformed cylinders, are also considered.


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Mathematical Notes, 2004, 75:3, 331–340

Bibliographic databases:

UDC: 517.958 517.95
Received: 26.05.2003

Citation: V. V. Grushin, “On the Eigenvalues of Finitely Perturbed Laplace Operators in Infinite Cylindrical Domains”, Mat. Zametki, 75:3 (2004), 360–371; Math. Notes, 75:3 (2004), 331–340

Citation in format AMSBIB
\by V.~V.~Grushin
\paper On the Eigenvalues of Finitely Perturbed Laplace Operators in Infinite Cylindrical Domains
\jour Mat. Zametki
\yr 2004
\vol 75
\issue 3
\pages 360--371
\jour Math. Notes
\yr 2004
\vol 75
\issue 3
\pages 331--340

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    This publication is cited in the following articles:
    1. V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations”, Theoret. and Math. Phys., 141:2 (2004), 1562–1592  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder”, Math. Notes, 77:5 (2005), 606–613  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. R. R. Gadyl'shin, “Local Perturbations of Quantum Waveguides”, Theoret. and Math. Phys., 145:3 (2005), 1678–1690  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Planida M. Yu., “Singular perturbation of the Dirichlet problem in an infinite cylinder”, Dokl. Math., 71:3 (2005), 466–469  mathscinet  isi  elib
    5. Bruening J., Dobrokhotov S., Sekerzh-Zenkovich S., Tudorovskiy T., “Spectral series of the Schrodinger operator in thin waveguides with periodic structure. I. Adiabatic approximation and semiclassical asymptotics in the 2D case”, Russ. J. Math. Phys., 13:4 (2006), 380–396  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. 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
    7. 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
    8. Ekholm T., Kovařík H., Krejčiřík D., “A Hardy inequality in twisted waveguides”, Arch. Ration. Mech. Anal., 188:2 (2008), 245–264  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. Krejcirik D., “Twisting Versus Bending in Quantum Waveguides”, Analysis on Graphs and its Applications, Proceedings of Symposia in Pure Mathematics, 77, ed. Exner P. Keating J. Kuchment P. Sunada T. Teplyaev A., Amer Mathematical Soc, 2008, 617–636  crossref  mathscinet  zmath  isi
    10. 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
    11. 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
    12. 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
    13. Borisov D. Cardone G., “Planar Waveguide with “Twisted” Boundary Conditions: Discrete Spectrum”, J. Math. Phys., 52:12 (2011), 123513  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    14. 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
    15. 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
    16. Briet Ph., Kovarik H., Raikov G., “Scattering in Twisted Waveguides”, J. Funct. Anal., 266:1 (2014), 1–35  crossref  mathscinet  zmath  isi  scopus  scopus
    17. 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
    18. 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
    19. 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
    20. Exner P. Kovarik H., Quantum Waveguides, Theoretical and Mathematical Physics, Springer-Verlag Berlin, 2015, 1–382  crossref  mathscinet  isi
    21. 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
    22. 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
    23. 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
    24. 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
    25. 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
    26. Bruneau V., Miranda P., Popoff N., “Resonances Near Thresholds in Slightly Twisted Waveguides”, Proc. Amer. Math. Soc., 146:11 (2018), 4801–4812  crossref  mathscinet  zmath  isi  scopus
    27. 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
    28. 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
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