Journal of Statistical Physics

, Volume 61, Issue 3–4, pp 869–876 | Cite as

The energy level spacing for two harmonic oscillators with golden mean ratio of frequencies

  • P. M. Bleher
Short Communications


Berry and Tabor discussed, among other things, a beautiful problem about the energy level spacing distribution for a system of two harmonic oscillators. They gave some interesting theoretical arguments which show that there is no level clustering for generic harmonic oscillators, and various numerical experiments were exposed and discussed. But the main question they posed about the existence of the limit distribution of the level spacing remained open. The present paper discusses this question in the case when the ratio of the frequencies is the golden mean σ=(√5−1)/2. The approach enables one to study the generic case of the frequency ratio as well, which is done elsewhere.

Key words

Quantum linear oscillators distribution of energy level spacing 


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  1. 1.
    M. V. Berry and M. Tabor, Level clustering in the regular spectrum,Proc. R. Soc. Lond. A 356:375–394 (1977).Google Scholar
  2. 2.
    A. Pandey, O. Bohigas and M. J. Giannoni, Level repulsion in the spectrum of two- dimensional harmonic oscillators,J. Phys. A. Math. Gen. 22, 4083–4088 (1989).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • P. M. Bleher
    • 1
    • 2
  1. 1.Institute for Applied MathematicsUSSR Academy of SciencesMoscowUSSR
  2. 2.Centre de Physique ThéoriqueCNRS Luminy Case 907Marseille Cedex 09France

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