Quantum chaotic dynamics and random polynomials

Journal Article · Sun Dec 01 04:00:00 EST 1996 · Journal of Statistical Physics

DOI:https://doi.org/10.1007/BF02199359· OSTI ID:539379

Bogomolny, E; Bohigas, O; Leboeuf, P ^[1]

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1. Institut de Physique Nucleaire, Orsay (France)

We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of {open_quotes}quantum chaotic dynamics.{close_quotes} 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.

• Sponsoring Organization: USDOE
• OSTI ID: 539379
• Journal Information: Journal of Statistical Physics, Journal Name: Journal of Statistical Physics Journal Issue: 5-6 Vol. 85; ISSN JSTPBS; ISSN 0022-4715
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
CORRELATION FUNCTIONS
DISTRIBUTION FUNCTIONS
ERGODIC HYPOTHESIS
HAMILTONIANS
POLYNOMIALS
QUANTUM MECHANICS
RANDOMNESS
SEMICLASSICAL APPROXIMATION