On the Applicability of the Energy Level Dynamics for the Hamiltonian Systems in the Transition Region between Integrability and Chaos

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Abstract
We study the energy level dynamics in the Dyson-Pechukas-Yukawa picture 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 "energy" and the square of the "angular momentum", which are the only two constants of motion quadratic in perturbation matrix elements. This fact implies the maximum-entropy property of the underlying canonical distribution, which thus makes the Yukawa joint distribution the most probable one. The resulting reduced statistics (a one-parameter family) is expected to provide significant global theoretical description in the quasi-universal non-semi-classical regime of finite typically observed in case of soft chaos (in the transition region between integrability and chaos). However, the power law level repulsion at small spacings cannot be adequately described, since one observes the linear level repulsion instead (or quadratic if there is no antiunitary symmetry), except possibly in the limit of infinitely many levels.

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