Journal of Statistical Physics

, Volume 85, Issue 5–6, pp 639–679 | Cite as

Quantum chaotic dynamics and random polynomials

  • E. Bogomolny
  • O. Bohigas
  • P. Leboeuf


We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of “quantum chaotic dynamics.” It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of self-inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials.

Key Words

random polynomials chaotic dynamics quantum mechanics semiclassical approximations random matrix theory 


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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • E. Bogomolny
    • 1
  • O. Bohigas
    • 1
  • P. Leboeuf
    • 1
  1. 1.Division de Physique Théorique (Unité de recherche des Universités de Paris XI et Paris VI associée au CNRS)Institut de Physique NucléaireOrsay CedexFrance

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