Analyses Statistics Character of Fluctuations in Degenerated Spectra

The degeneracy appears generally in the energy level spectra. It is related closely to the symmetrical character of the corresponding Hamiltonian random matrix. According to the fact that the state space of Hamiltonian matrix for degenerated spectra can be expanded into a series of sub-space corresponding to different eigenvalues, the degenerated spectra is dealed with by a special way. By introducing weights of the levels, the accumulative function curve of the degenerated spectra is smoothed. Therefore the fluctuation spectra can be unfolded by means of polynomial expansion fitting with weights. Then a set of methods that are used to analyses statistical character of fluctuation such as the NNS distribution, the spectra rigidity and the fractal dimension function for energy levels are suggested. Further more, the degenerated spectra such as the vibrational energy spectra of the molecules of H2O、 NH3、 CH4 are analyzed with this method. It's tums out that both their regular spectra and reduced non-degenerated spectra are nonpoissonized, further more, the former is obviously further nonpoissonized compared with the latter,i.e.,the level repulsion of the reduced non-degenerated spectra is greater than the corresponding regular one. However, the statistic character of fluctuations in energy spectra of the H2O, NH3, CH4 are different because of their degeneracies: they transform from the obviously nonpoissonized type to the closely poisson type.