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TMF, 2004, Volume 141, Number 2, Pages 267–303 (Mi tmf120)  

This article is cited in 30 scientific papers (total in 30 papers)

Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

V. V. Belova, S. Yu. Dobrokhotovb, T. Ya. Tudorovskiib

a Moscow State Institute of Electronics and Mathematics (Technical University)
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: We consider equations of nonrelativistic quantum mechanics in thin three-dimensional tubes (nanotubes). We suggest a version of the adiabatic approximation that permits reducing the original three-dimensional equations to one-dimensional equations for a wide range of energies of longitudinal motion. The suggested reduction method (the operator method for separating the variables) is based on the Maslov operator method. We classify the solutions of the reduced one-dimensional equation. In Part I of this paper, we deal with the reduction problem, consider the main ideas of the operator separation of variables (in the adiabatic approximation), and derive the reduced equations. In Part II, we will discuss various asymptotic solutions and several effects described by these solutions.

Keywords: nanotubes, adiabatic approximation, size quantization, spin-orbit interaction, semiclassical approximation

DOI: https://doi.org/10.4213/tmf120

Full text: PDF file (567 kB)
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English version:
Theoretical and Mathematical Physics, 2004, 141:2, 1562–1592

Bibliographic databases:

Received: 22.09.2003
Revised: 28.04.2004

Citation: V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations”, TMF, 141:2 (2004), 267–303; Theoret. and Math. Phys., 141:2 (2004), 1562–1592

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. V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder”, Math. Notes, 77:5 (2005), 606–613  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    3. V. V. Belov, S. Yu. Dobrokhotov, V. P. Maslov, T. Ya. Tudorovskii, “A generalized adiabatic principle for electron dynamics in curved nanostructures”, Phys. Usp., 48:9 (2005), 962–968  mathnet  crossref  crossref  adsnasa  isi  elib  elib
    4. A. V. Krivko, V. V. Kucherenko, “Semiclassical Asymptotics of the Matrix Sturm–Liouville Problem”, Math. Notes, 80:1 (2006), 136–140  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Belov VV, Dobrokhotov SY, Tudorovskiy TY, “Operator separation of variables for adiabatic problems in quantum and wave mechanics”, Journal of Engineering Mathematics, 55:1–4 (2006), 183–237  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. Dell'Antonio G, Tenuta L, “Quantum graphs as holonomic constraints”, Journal of Mathematical Physics, 47:7 (2006), 072102  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Kryvko A, Kucherenko VV, “Semiclassical asymptotics of the vector Sturm-Liouville problem”, Russian Journal of Mathematical Physics, 13:2 (2006), 188–202  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Bruening J, Dobrokhotov S, Sekerzh-Zenkovich S, et al, “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
    9. Kryvko A., Kucherenko V.V., “Quantization conditions for the vector Sturm-Liouville problem”, Proceedings of the International Conference Days on Diffraction 2006, 2006, 117–125  crossref  isi  scopus  scopus
    10. 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
    11. Dell'Antonio, GF, “Dynamics on quantum graphs as constrained systems”, Reports on Mathematical Physics, 59:3 (2007), 267  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. Belov, VV, “Integrable models of the longitudinal motion of electrons in curved 3D nanotubes”, Doklady Mathematics, 75:1 (2007), 147  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
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    14. 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
    15. J. Brüning, S. Yu. Dobrokhotov, R. V. Nekrasov, A. I. Shafarevich, “Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity”, Theoret. and Math. Phys., 155:2 (2008), 689–707  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    16. Bruning, J, “Quantum dynamics in a thin film, I. Propagation of localized perturbations”, Russian Journal of Mathematical Physics, 15:1 (2008), 1  crossref  mathscinet  adsnasa  isi
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    18. 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
    19. Zalipaev, VV, “The Gaussian beams summation method in the quantum problems of electronic motion in a magnetic field”, Journal of Physics A-Mathematical and Theoretical, 42:20 (2009), 205302  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    20. Wachsmuth J., Teufel S., “Constrained quantum systems as an adiabatic problem”, Phys Rev A, 82:2 (2010), 022112  crossref  adsnasa  isi  elib  scopus  scopus
    21. J. Brüning, V. V. Grushin, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Generalized Foldy–Wouthuysen transformation and pseudodifferential operators”, Theoret. and Math. Phys., 167:2 (2011), 547–566  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    22. Bruening J., Dobrokhotov S.Yu., Sekerzh-Zen'kovich S.Ya., Tudorovskiy T.Ya., “Spectral Series of the Schrodinger Operator in a Thin Waveguide with a Periodic Structure. 2. Closed Three-Dimensional Waveguide in a Magnetic Field”, Russian Journal of Mathematical Physics, 18:1 (2011), 33–53  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
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    25. J. Brüning, S. Yu. Dobrokhotov, R. V. Nekrasov, “Splitting of lower energy levels in a quantum double well in a magnetic field and tunneling of wave packets in nanowires”, Theoret. and Math. Phys., 175:2 (2013), 620–636  mathnet  crossref  crossref  zmath  adsnasa  isi  elib  elib
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    27. J. Brüning, S. Yu. Dobrokhotov, M. I. Katsnel'son, D. S. Minenkov, “Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems”, Theoret. and Math. Phys., 186:3 (2016), 333–345  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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