Chiyoda-ku, Tokyo 101-8308, Japan

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 [1]. In 1985,
Yukawa [2] gave an excellent idea to construct such a one-parameter
family of -level distributions, based on the so-called level
dynamics [3]: 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

Here, the parameter that appears is which interpolates the two limits as

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 [4]: he showed that for this
class expression (1) can be rewritten as a determinant

where the right-hand side allows an infinite sum over with coefficient , called the grand canonical series with fugacity . In the present talk, it will be extended to the other two classes i.e. orthogonal and symplectic classes by using

**References**

[1] For example, A.D. Mirlin, Phys. Rep. **326**(2000), 259.

[2] T. Yukawa, Phys. Rev. Lett. **54**(1985), 1883; Phys. Lett.
**A 116**(1986), 227; **A 245**(1998), 183.

[3] F. Haake, *Quantum Signature of Chaos*, Springer Verlag, Berlin,
New York (1991).

[4] P.J. Forrester, Phys. Lett. **A 173**(1993), 355.

[5] F.J. Dyson, Comm. Math. Phys. **19**(1970), 235.

[6] H. Hasegawa and Y. Sakamoto, *Grand canonical random matrices for
intermediate level statistics*, Phys. Lett. **A 297**(2002), 146.