Abstract: Through studying the quantum counterparts of two kinds of classical conservative systems, such as Henon-Heiles model, Barbains model and the model of the rotating-ball a-long a rigid ring, this title explains the connections of statistical characteristics of the fluctuations in energy spectra with the dynamic behavior of these models. Then, as a result, it reveals that there are two kinds of expressions of quantum chaos in the statistical characteristics of spectral fluctuations. Furthermore, by hamiltonizing the evolution equation of Lotka-Volterra model which belongs to classical dissipative systems in principle, we find that the statistical characteristics of the fluctuations in eigenvalue-spectra of its quantum counterpart go completely beyond the frame of Poisson-GOE(Winger) or GUE distribution. It shows that the fluctuations in eigenvalue-spectra of the quantum counterparts of the classical dissipative and the conservative systems follow different statistical regularities respectively.

Key words: Quantum chaos, Eigenvalue-spectra, Spectral fluctuation statistics