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Theoretical and Mathematical Physics

, Volume 175, Issue 2, pp 620–636 | Cite as

Splitting of lower energy levels in a quantum double well in a magnetic field and tunneling of wave packets in Nanowires

  • J. Brüning
  • S. Yu. Dobrokhotov
  • R. V. Nekrasov
Article

Abstract

We consider the problem of the splitting of lower eigenvalues of the two-dimensional Schrödinger operator with a double-well-type potential in the presence of a homogeneous magnetic field. The main result is the observation that the partial Fourier transformation takes the operator under study to a Schrödingertype operator with a (new) double-well-type potential but already without any magnetic field. We use this observation to investigate the influence of the magnetic field on the tunneling effects. We discuss two methods for calculating the splitting of lower eigenvalues: based on the instanton and based on the so-called libration. We use the obtained result to study the tunneling of wave packets in parallel quantum nanowires in a constant magnetic field.

Keywords

Schrödinger operator double-well-type potential homogeneous magnetic field tunneling double quantum wire wave packet 

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References

  1. 1.
    V. P. Maslov, Proc. Steklov Inst. Math., 163, 177–209 (1985); A. M. Polyakov, Nucl. Phys. B, 120, 429–458 (1977); E. M. Harrell, Commun. Math. Phys., 75, 239–261 (1980); B. Helffer and J. Sjöstrand, Commun. Partial Differ. Equations, 9, 337–408 (1984); Ann. Inst. H. Poincaré Phys. Théor., 42, 127–212 (1985); S. R. Coleman, “The uses of instantons,” in: TheWhys of Subnuclear Physics (Erice, Italy, 23 July–10 August 1977, A. Zichichi, ed.), Plenum, New York (1979), pp. 805–916; B. Simon, Ann. Inst. H. Poincaré Sec. A, n.s., 38, 295–308 (1983); Ann. Math. (2), 120, 89–118 (1984); S. C. Creagh, J. Phys. A, 27, 4969–4993 (1994); S. Yu. Dobrokhotov, V. N. Kolokoltsov, and V. P. Maslov, Theor. Math. Phys., 87, 561–599 (1991); “Quantization of the Bellman equation, exponential asymptotics, and tunneling,” in: Idempotent Analysis (Adv. in Sov. Math., Vol. 13), Amer. Math. Soc., Providence, R. I. (1992), pp. 1–46.zbMATHGoogle Scholar
  2. 2.
    S. Yu. Dobrokhotov and V. N. Kolokoltsov, Theor. Math. Phys., 94, 300–305 (1993); S. Yu. Dobrokhotov and V. N. Kolokoltsov, J. Math. Phys., 36, 1038–1053 (1995).MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Brüning, S. Yu. Dobrokhotov, and E. S. Semenov, Regul. Chaotic Dyn., 11, 167–180 (2006).MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    A. Yu. Anikin, Russ. J. Math. Phys., 20, 1–10 (2013); A. Yu. Anikin, Theor. Math. Phys., 175, 609–619 (2013).Google Scholar
  5. 5.
    V. V. Kozlov, J. Appl. Math. Mech., 40, 363–370 (1976); S. V. Bolotin and, V. V. Kozlov, J. Appl. Math. Mech., 42, 256–261 (1978).zbMATHCrossRefGoogle Scholar
  6. 6.
    V. F. Gantmakher and Y. B. Levinson, Carrier Scattering in Metals and Semiconductors, North-Holland, Amsterdam (1987); L. I. Magarill and M. V. Éntin, JETP, 96, 766–774 (2003); M. V. Entin and L. I. Magarill, Phys. Rev. B, 66, 205308 (2002); A. I. Vedernikov and A. V. Chaplik, JETP, 90, 397–399 (2000); V. V. Belov, S. Yu. Dobrokhotov, and V. P. Maslov, Theor. Math. Phys., 141, 1562–1592 (2004); V. V. Belov, S. Yu. Dobrokhotov, V. P. Maslov, and T. Ya. Tudorovskii, Phys. Usp., 48, 962–968 (2005).Google Scholar
  7. 7.
    V. P. Maslov, The Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow (1977); English transl. (Progr. Phys., Vol. 16), Birkhäuser, Basel (1994).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • J. Brüning
    • 1
  • S. Yu. Dobrokhotov
    • 2
    • 3
  • R. V. Nekrasov
    • 2
    • 3
  1. 1.Humboldt UniversityBerlinGermany
  2. 2.Ishlinskiy Institute for Problems in MechanicsMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyMoscowRussia

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