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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


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English version:
Theoretical and Mathematical Physics, 2004, 141:2, 1562–1592

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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
\by V.~V.~Belov, S.~Yu.~Dobrokhotov, T.~Ya.~Tudorovskii
\paper Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I.~Reduction to~Spatially One-Dimensional Equations
\jour TMF
\yr 2004
\vol 141
\issue 2
\pages 267--303
\jour Theoret. and Math. Phys.
\yr 2004
\vol 141
\issue 2
\pages 1562--1592

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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
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    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
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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