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Mat. Zametki, 2005, Volume 77, Issue 5, Pages 656–664 (Mi mz2524)  

This article is cited in 10 scientific papers (total in 10 papers)

Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: In this paper, we derive sufficient conditions for the existence of an eigenvalue for the Laplace and the Schrödinger operators with transversal potential for homogeneous Dirichlet boundary conditions in a tube, i.e., in a curved and twisted infinite cylinder. For tubes with small curvature and small internal torsion, we derive an asymptotic formula for the eigenvalue of the problem. We show that, under certain relations between the curvature and the internal torsion of the tube, the above operators possess no discrete spectrum.

DOI: https://doi.org/10.4213/mzm2524

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English version:
Mathematical Notes, 2005, 77:5, 606–613

Bibliographic databases:

UDC: 517.958
Received: 28.04.2004
Revised: 23.09.2004

Citation: V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder”, Mat. Zametki, 77:5 (2005), 656–664; Math. Notes, 77:5 (2005), 606–613

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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”, Russian Journal of Mathematical Physics, 13:4 (2006), 380–396  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. 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
    3. Kovařik H., Sacchetti A., “Resonances in twisted quantum waveguides”, J. Phys. A, 40:29 (2007), 8371–8384  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. 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
    5. 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  scopus
    6. Krejcirik D., “Twisting Versus Bending in Quantum Waveguides”, Analysis on Graphs and its Applications, Proceedings of Symposia in Pure Mathematics, 77, eds. Exner P., Keating J., Kuchment P., Sunada T., Teplyaev A., Amer Mathematical Soc, 2008, 617–636  crossref  mathscinet  zmath  isi
    7. 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
    8. 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
    9. Exner P. Kovarik H., Quantum Waveguides, Theoretical and Mathematical Physics, Springer-Verlag Berlin, 2015, 1–382  crossref  mathscinet  isi
    10. 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
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