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

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Matematicheskie Zametki, 2008, Volume 83, Issue 4, Pages 503–519
DOI: https://doi.org/10.4213/mzm4573 (Mi mzm4573)

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

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

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Full-text PDF (604 kB) Citations (9)

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

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.

Received: 17.09.2007

English version:
Mathematical Notes, 2008, Volume 83, Issue 4, Pages 463–477
DOI: https://doi.org/10.1134/S000143460803019X

Bibliographic databases:

UDC: 517.958

Language: Russian

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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This publication is cited in the following 9 articles:

Haag S., Lampart J., Teufel S., “Quantum Waveguides With Magnetic Fields”, Rev. Math. Phys., 31:8 (2019), 1950025
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
Tusek M., “On an extension of the Iwatsuka model”, J. Phys. A-Math. Theor., 49:36 (2016), 365205
D.I. Borisov, “The Emergence of Eigenvalues of a $\mathcal{PT}$ -Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883
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
Krejcirik D., Raymond N., “Magnetic Effects in Curved Quantum Waveguides”, Ann. Henri Poincare, 15:10 (2014), 1993–2024
Stockhofe J., Schmelcher P., “Nonadiabatic Couplings and Gauge-Theoretical Structure of Curved Quantum Waveguides”, Phys. Rev. A, 89:3 (2014), 033630
Borisov D. Cardone G., “Planar waveguide with “twisted” boundary conditions: small width”, J. Math. Phys., 53:2 (2012), 023503, 22 pp.
V. V. Grushin, “Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions”, Math. Notes, 86:6 (2009), 767–774

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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References:	116
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