Abstract
The statistical properties of random analytic functions ψ(z) are investigated as a phase-space model for eigenfunctions of fully chaotic systems. We generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of ψ(z) previously demonstrated for a spherical phase space [SU(2) polynomials]. For systems with time-reversal symmetry, the number of real roots is computed for the three geometries. In the semiclassical regime, the local correlation functions are shown to be universal, independent of the system considered or the geometry of phase space. In particular, the autocorrelation function of ψ is given by a Gaussian function. The connections between this model and the Gaussian random function hypothesis as well as the random matrix theory are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. L. Mehta, Random Matrices (Academic Press, New York, 1991).
O. Bohigas, in Proceedings of the Les Houches Summer School, Chaos et Physique Quantique/Chaos and Quantum Physics, Les Houches, Session LII, M.-J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (Elsevier, 1991).
C. E. Porter, ed., Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965).
A. Voros, Ann. Inst. Henri Poincaré A 24:31 (1976); Ann. Inst. Henri Poincaré A 26:343 (1977).
M. V. Berry, J. Phys. A: Math. Gen. 10:2083 (1977).
E. B. Bogomolny, Physica D 31:169 (1988).
M. V. Berry, Proc. R. Soc. London A 423:219 (1989).
E. Bogomolny, O. Bohigas, and P. Lebœuf, Phys. Rev. Lett. 68:2726 (1992).
E. Bogomolny, O. Bohigas, and P. Lebœuf, J. Stat. Phys. 85:639 (1996).
B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, preprint math_9803052.
N. L. Balazs and A. Voros, Phys. Rep. 143:109 (1986).
A. Perelomov, Generalized Coherent States and their Applications (Springer, New York, 1986).
V. I. Bargmann, Commun. Pure Appl. Math. 14:187 (1961); 20:1 (1967).
A. Edelman and E. Kostlan, Bull. Amer. Math. Soc. 32:1 (1995).
J. Hannay, J. Phys. A: Math. Gen. 29:L101 (1996).
P. Lebœuf and P. Shukla, J. Phys. A: Math. Gen. 29:4827 (1996).
P. Bleher and X. Di, J. Stat. Phys. 88:269 (1997).
M. Kac, Probability and Related Topics in Physical Sciences (Wiley, New York, 1959).
J. Hannay, The chaotic analytic function (preprint 1998).
The observation-based on numerical computations-that the zeros of chaotic eigenfunctions tend to fill uniformly the whole available phase space was made in P. Lebœuf and A. Voros, J. Phys. A: Math. Gen. 23:1765 (1990); see also Quantum Chaos, G. Casati and B. Chirikov, eds. (Cambridge University Press, 1995).
S. Nonnenmacher and A. Voros, J. Stat. Phys. 92:431 (1998).
T. Prosen, Physica D 91:244 (1996) and private communication.
J.-M. Tualle and A. Voros, Chaos, Solitons & Fractals 5:1085 (1995).
T. Prosen, J. Phys. A 29:4417 (1996).
V. N. Prigodin et al., Phys. Rev. Lett. 75:2392 (1995); see also a comment on the previous paper by M. Srednicki, cond-mat/9512115.
P. Forrester and G. Honner, “ Exact statistical properties of the zeros of complex random polynomials,” preprint 1998.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Leboeuf, P. Random Analytic Chaotic Eigenstates. Journal of Statistical Physics 95, 651–664 (1999). https://doi.org/10.1023/A:1004595310043
Issue Date:
DOI: https://doi.org/10.1023/A:1004595310043