ON THE APPLICABILITY OF THE ENERGY-LEVEL DYNAMICS FOR THE HAMILTONIAN-SYSTEMS IN THE TRANSITION REGION BETWEEN INTEGRABILITY AND CHAOS

We study the energy level dynamics in the Dyson-Pechukas-Yukawa pictur e aiming at the understanding of the statistical properties of energy spectra of generic Hamiltonian systems between integrability and chaos . We discuss the role of the major integrals of motion, namely the <<e nergy>> and the square of the <<angular momentum>>, which are the only two constants of motion quadratic in perturbation matrix elements. Th is fact implies the maximum-entropy property of the underlying canonic al distribution, which thus makes the Yukawa joint distribution the mo st probable one. The resulting reduced statistics (a one-parameter fam ily) is expected to provide significant global theoretical description in the quasi-universal non-semi-classical regime of finite hBAR typic ally observed in case of soft chaos (in the transition region between integrability and chaos). However, the power law level repulsion at sm all spacings cannot be adequately described, since one observes the li near level repulsion instead (or quadratic if there is no antiunitary symmetry), except possibly in the limit of infinitely many levels.