There has been a long-standing subject in quantum-chaos studies to
achieve a framework of energy-level statistics which interplolates
between Poisson for integrable systems and those of Gaussian
random matrix ensembles (GOE, GUE and
GSE) for fully chaotic systems by means of a parameter . In 1985,
Yukawa  gave an excellent idea to construct such a one-parameter
family of -level distributions, based on the so-called level
dynamics : one considers an Hermitian matrix of the
form (a perturbation of by another with
perturbation strength which is regarded as the``time''). His
result can be summarized by a special form
A further treatment of the distribution (1) to deduce concrete
statistics (the spacing distribution, long-range 2-level correlations,
etc.) is a hard problem, and have so far produced little
fruits. However, for the unitary symmetry class , a fully
analytic method was devised by Forrester : he showed that for this
class expression (1) can be rewritten as a determinant
 For example, A.D. Mirlin, Phys. Rep. 326(2000), 259.
 T. Yukawa, Phys. Rev. Lett. 54(1985), 1883; Phys. Lett. A 116(1986), 227; A 245(1998), 183.
 F. Haake, Quantum Signature of Chaos, Springer Verlag, Berlin, New York (1991).
 P.J. Forrester, Phys. Lett. A 173(1993), 355.
 F.J. Dyson, Comm. Math. Phys. 19(1970), 235.
 H. Hasegawa and Y. Sakamoto, Grand canonical random matrices for intermediate level statistics, Phys. Lett. A 297(2002), 146.