Non-Separable Variables: Hierarchical Quantization and Tunneling Resonances

  • Mark Ya. Azbel
Part of the NATO ASI Series book series (NSSB, volume 254)


The study of an electron in a co-focal elliptic shell is very instructive. In classical mechanics its motion is chaotic. In quantum mechanics variables are separable in the absence of magnetic field. Then the energy spectrum is readily determined. When the interfocal distance d infinitely increases, while the shell width is kept finite, the spectrum above a mobility edge energy E n consists of intersecting branches (n is the branch number). Lower energy states E < E n are localized in the widest region [1].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M.Ya. Azbel and O. Entin-Wohlman, J. Phys. A22, L957 (1989); Phys. Rev. B 41, 395 (1990)ADSGoogle Scholar
  2. [2]
    D. Bergman, O. Entin-Wohlman, and M.Ya. Azbel, to be publishedGoogle Scholar
  3. [3]
    B.B. Suprapto and P.B. Butcher, J. Phys. C8, L517 (1975);ADSGoogle Scholar
  4. [3a]
    B.I. Shklovskii and A.L. Efros, Electronic properties of doped semiconductors 45, 210, Springer-Verlag, New York (1984)CrossRefGoogle Scholar
  5. [4]
    A. Hartstein, A.B. Fowler, and K.C. Woo, Physica 117–118B, 655 (1983); for later experiments see Y.Google Scholar
  6. [4a]
    Shapir and Z. Ovadyahu, Phys. Rev. B40, 12441 (1989) and refs. thereinCrossRefGoogle Scholar
  7. [5]
    V.I. Nguyen, B.Z. Spivak, B.I. Shklovskii, JETP Lett. 41, 42 (1985)ADSGoogle Scholar
  8. [5a]
    V.I. Nguyen, B.Z. Spivak, B.I. Shklovskii, Sov. Phys. JETP 62, 1021 (1982)Google Scholar
  9. [6]
    O. Entin-Wohlman, Y. Imry, and V. Sivan, Phys. Rev. B40, 8342 (1989) and refs. thereinADSCrossRefGoogle Scholar
  10. [7]
    R.F. Kazarinov and S. Luryi, Phys. Rev. B25, 7626 (1982)ADSCrossRefGoogle Scholar
  11. [7a]
    R.F. Kazarinov and S. Luryi, Phys. Rev. B27, 1386 (1983)CrossRefMathSciNetGoogle Scholar
  12. [8]
    M.Ya. Azbel, Sol. St. Commun. 54, 127 (1985);ADSCrossRefGoogle Scholar
  13. [8a]
    M.Ya. Azbel and O. Entin-Wohlman, Phys. Rev. B32, 562 (1985)ADSCrossRefGoogle Scholar
  14. [9]
    B.I. Halperin, Helv. Phys. Acta 56, 75 (1983);Google Scholar
  15. [9a]
    B.I. Halperin, Phys. Rev. B25, 2185 (1982);ADSCrossRefMathSciNetGoogle Scholar
  16. [9b]
    M.Ya. Azbel and M.H. Brodsky, Phil. Mag. 50, 237 (1984)CrossRefGoogle Scholar
  17. [10]
    M.Ya. Azbel, Phys. Rev. Lett. 47, 1015 (1981)ADSCrossRefGoogle Scholar
  18. [11]
    I.M. Lifshitz, Usp. Fiz. Nauk 83, 617 (1964)Google Scholar
  19. [12]
    It is convenient to chose V(y) = (V 0/y)sin(y/y*). Then W(k) = V 0 for |k| < 1/y*; W(k) = 0 for |k| > 1/y*, and a single impurity yields K = (y* sinh V 0 -1)-1 . The Lifshitz model is valid (K ≪ 1/y*) in shallow well: V 0 ≪ 1.Google Scholar
  20. [13]
    A. Miller and E. Abrahams, Phys. Rev. 120, 745 (1960);ADSCrossRefzbMATHGoogle Scholar
  21. [13a]
    V. Ambegaokar, B.I. Halperin, and J.S. Langer, Phys. Rev. B4, 2612 (1971), and ref. [3]ADSCrossRefGoogle Scholar
  22. [14]
    See on their nature R. Meir and A. Aharony, Phys. Rev. B37, 6349 (1989), and refs. thereinCrossRefGoogle Scholar
  23. [15]
    See, e.g., M.Ya. Azbel, Sol. St. Commun. 43, 515 (1982); Topics in Sol. St. Phys. 61, 162 (1985)ADSCrossRefGoogle Scholar
  24. [16]
    See, e.g., V. MacColl, Phys. Rev. 40, 621 (1932)ADSCrossRefGoogle Scholar
  25. [16a]
    E.P. Wigner, Phys. Rev. 98, 145 (1955);ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. [16b]
    A.J. Leggett, Prog. Theor. Phys. Suppl. 69, 80 (1980).ADSCrossRefMathSciNetGoogle Scholar
  27. [16c]
    E.H. Hauge and J.A. Stovneng, Rev. Mod. Phys. 61, 917 (1989), and refs. thereinADSCrossRefGoogle Scholar
  28. [17]
    M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982)ADSCrossRefGoogle Scholar
  29. [17a]
    M. Büttiker and R. Landauer, Phys. Scripta 32, 429 (1987);CrossRefGoogle Scholar
  30. [17b]
    M. Büttiker and R. Landauer, IBM J. of Res. and Devel. 30, 451 (1986);CrossRefGoogle Scholar
  31. [17c]
    R. Landauer, Nature 341, 567 (1989). See a detailed discussion on tunneling traversal time and refs. in M. Büttiker, to be publishedADSCrossRefGoogle Scholar
  32. [18]
    Note that an arbitrarily large (albeit exponentially weak) energy increase allows for the arbitrarily quick (albeit exponentially weak) response at any distance x Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Mark Ya. Azbel
    • 1
  1. 1.Raymond and Beverly Sackler Faculty of Exact Sciences School of Physics and AstronomyTel Aviv University Ramat AvivTel AvivIsrael

Personalised recommendations