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

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
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
\Bibitem{Gru05} \by V.~V.~Grushin \paper Asymptotic Behavior of the Eigenvalues of the Schr\"odinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder \jour Mat. Zametki \yr 2005 \vol 77 \issue 5 \pages 656--664 \mathnet{http://mi.mathnet.ru/mz2524} \crossref{https://doi.org/10.4213/mzm2524} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2178837} \zmath{https://zbmath.org/?q=an:1082.35113} \elib{http://elibrary.ru/item.asp?id=9155817} \transl \jour Math. Notes \yr 2005 \vol 77 \issue 5 \pages 606--613 \crossref{https://doi.org/10.1007/s11006-005-0062-7} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000230336000002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-21644478248} 

• http://mi.mathnet.ru/eng/mz2524
• https://doi.org/10.4213/mzm2524
• http://mi.mathnet.ru/eng/mz/v77/i5/p656

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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. 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
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
3. Kovařik H., Sacchetti A., “Resonances in twisted quantum waveguides”, J. Phys. A, 40:29 (2007), 8371–8384
4. V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator in Thin Closed Tubes”, Math. Notes, 83:4 (2008), 463–477
5. Ekholm T., Kovařík H., Krejčiřík D., “A Hardy inequality in twisted waveguides”, Arch. Ration. Mech. Anal., 188:2 (2008), 245–264
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
7. V. V. Grushin, “Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes”, Math. Notes, 85:5 (2009), 661–673
8. Borisov D., Cardone G., “Planar Waveguide with “Twisted” Boundary Conditions: Discrete Spectrum”, J. Math. Phys., 52:12 (2011), 123513
9. Exner P. Kovarik H., Quantum Waveguides, Theoretical and Mathematical Physics, Springer-Verlag Berlin, 2015, 1–382
10. Bikmetov A.R. Gadyl'shin R.R., “On local perturbations of waveguides”, Russ. J. Math. Phys., 23:1 (2016), 1–18
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