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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

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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
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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. 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
2. T. Ya. Tudorovskii, “On the Effect of Spin on Classical and Quantum Dynamics of an Electron in Thin Twisted Tubes”, Math. Notes, 78:6 (2005), 883–889
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
4. A. V. Krivko, V. V. Kucherenko, “Semiclassical Asymptotics of the Matrix Sturm–Liouville Problem”, Math. Notes, 80:1 (2006), 136–140
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
6. Dell'Antonio G, Tenuta L, “Quantum graphs as holonomic constraints”, Journal of Mathematical Physics, 47:7 (2006), 072102
7. Kryvko A, Kucherenko VV, “Semiclassical asymptotics of the vector Sturm-Liouville problem”, Russian Journal of Mathematical Physics, 13:2 (2006), 188–202
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
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
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
11. Dell'Antonio, GF, “Dynamics on quantum graphs as constrained systems”, Reports on Mathematical Physics, 59:3 (2007), 267
12. Belov, VV, “Integrable models of the longitudinal motion of electrons in curved 3D nanotubes”, Doklady Mathematics, 75:1 (2007), 147
13. Bruening J., Grikurov V.E., “Electron scattering by nanotube Y-junction. Computation and application to modeling by graphs”, Days on Diffraction 2007, 2007, 31–37
14. V. V. Grushin, “Asymptotic Behavior of the Eigenvalues of the Schrödinger Operator in Thin Closed Tubes”, Math. Notes, 83:4 (2008), 463–477
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
16. Bruning, J, “Quantum dynamics in a thin film, I. Propagation of localized perturbations”, Russian Journal of Mathematical Physics, 15:1 (2008), 1
17. Bruning, J, “Numerical simulation of electron scattering by nanotube junctions”, Russian Journal of Mathematical Physics, 15:1 (2008), 17
18. V. V. Grushin, “Asymptotic Behavior of Eigenvalues of the Laplace Operator in Thin Infinite Tubes”, Math. Notes, 85:5 (2009), 661–673
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
20. Wachsmuth J., Teufel S., “Constrained quantum systems as an adiabatic problem”, Phys Rev A, 82:2 (2010), 022112
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
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
23. Borisov D., Cardone G., “Planar waveguide with “twisted” boundary conditions: Small width”, J Math Phys, 53:2 (2012), 023503
24. Zalipaev V.V., “High-Energy Localized Eigenstates in a Fabry-Perot Graphene Resonator in a Magnetic Field”, J. Phys. A-Math. Theor., 45:21 (2012), 215306
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
26. D.I. Borisov, “The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883
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
28. Streubel R., Fischer P., Kronast F., Kravchuk V.P., Sheka D.D., Gaididei Yu., Schmidt O.G., Makarov D., “Magnetism in curved geometries”, J. Phys. D-Appl. Phys., 49:36 (2016), 363001
29. D. I. Borisov, M. Znojil, “On eigenvalues of a $\mathscr{PT}$-symmetric operator in a thin layer”, Sb. Math., 208:2 (2017), 173–199
30. Cruz Philip Christopher S., Bernardo Reginald Christian S., Esguerra Jose Perico H., “Energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in electric and magnetic fields”, Ann. Phys., 379 (2017), 159–174
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