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Mat. Zametki, 2009, Volume 85, Issue 5, Pages 687–701 (Mi mz3886)  

This article is cited in 9 scientific papers (total in 9 papers)

Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: In this paper, we obtain an asymptotic expansion for the eigenvalues of the Laplace operator with zero Dirichlet conditions in tubes, i.e., in infinite bent cylinders with internal torsion under uniform contraction of their cross-sections, with respect to a small parameter characterizing the transverse dimensions of the tube. A method of reducing the problem of determining the eigenvalues to the solution of an implicit equation is proposed.

Keywords: eigenvalues of the Laplace operator, Dirichlet condition, thin infinite tube, Frenet equations, Schrödinger operator, Maslov canonical operator, quantum waveguide

DOI: https://doi.org/10.4213/mzm3886

Full text: PDF file (552 kB)
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English version:
Mathematical Notes, 2009, 85:5, 661–673

Bibliographic databases:

UDC: 517.958
Received: 05.10.2006

Citation: V. V. Grushin, “Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes”, Mat. Zametki, 85:5 (2009), 687–701; Math. Notes, 85:5 (2009), 661–673

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. de Oliveira C.R., Verri A.A., “On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes”, J. Math. Anal. Appl., 381:1 (2011), 454–468  crossref  mathscinet  zmath  isi  scopus
    2. 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  scopus
    3. D. KrejčiřÍk, N. Raymond, “Magnetic effects in curved quantum waveguides”, Ann. Henri Poincaré, 15:10 (2014), 1993–2024  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. J. Stockhofe, P. Schmelcher, “Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides”, Phys. Rev. A, 89:3 (2014), 033630  crossref  adsnasa  isi  elib  scopus
    5. 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
    6. Exner P. Kovarik H., Quantum Waveguides, Theoretical and Mathematical Physics, Springer-Verlag Berlin, 2015, 1–382  crossref  mathscinet  isi
    7. D. I. Borisov, M. Znojil, “On eigenvalues of a $\mathscr{PT}$-symmetric operator in a thin layer”, Sb. Math., 208:2 (2017), 173–199  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. 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
    9. Raymond N., “Bound States of the Magnetic Schrodinger Operator”, Bound States of the Magnetic Schrodinger Operator, Ems Tracts in Mathematics, 27, Eur. Math. Soc., 2017, 1–380  crossref  mathscinet  isi
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