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Mat. Zametki, 2008, Volume 83, Issue 4, Pages 503–519 (Mi mz4573)  

This article is cited in 8 scientific papers (total in 8 papers)

Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator in Thin Closed Tubes

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: In the present paper, we obtain an asymptotic expansion of the eigenvalues of the Schrödinger operator with the magnetic field taken into account and with zero Dirichlet conditions in closed tubes, i.e., in closed curved cylinders with intrinsic torsion under uniform compression of the transverse cross-sections, with respect to a small parameter characterizing the tube's transverse dimensions. We propose a method for reducing the eigenvalue problem to the problem of solving an implicit equation.

Keywords: Schrödinger operator, eigenvalue problem, asymptotics, thin closed tube, small perturbation, Dirichlet condition, Laplace operator

DOI: https://doi.org/10.4213/mzm4573

Full text: PDF file (604 kB)
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English version:
Mathematical Notes, 2008, 83:4, 463–477

Bibliographic databases:

UDC: 517.958
Received: 17.09.2007

Citation: V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator in Thin Closed Tubes”, Mat. Zametki, 83:4 (2008), 503–519; Math. Notes, 83:4 (2008), 463–477

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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. V. V. Grushin, “Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions”, Math. Notes, 86:6 (2009), 767–774  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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. Bedoya R., de Oliveira C.R., Verri A.A., “Complex Gamma-Convergence and Magnetic Dirichlet Laplacian in Bounded Thin Tubes”, J. Spectr. Theory, 4:3 (2014), 621–642  crossref  mathscinet  zmath  isi  scopus
    4. Krejcirik D., Raymond N., “Magnetic Effects in Curved Quantum Waveguides”, Ann. Henri Poincare, 15:10 (2014), 1993–2024  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Stockhofe J., Schmelcher P., “Nonadiabatic Couplings and Gauge-Theoretical Structure of Curved Quantum Waveguides”, Phys. Rev. A, 89:3 (2014), 033630  crossref  adsnasa  isi  elib  scopus
    6. 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
    7. Tusek M., “On an extension of the Iwatsuka model”, J. Phys. A-Math. Theor., 49:36 (2016), 365205  crossref  mathscinet  zmath  isi  elib  scopus
    8. 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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