Unfolding of the Spectrum for Chaotic and Mixed Systems

Random Matrix Theory (RMT) is capable of making predictions for the spectral fluctuations of a physical system only after removing the influence of the level density by unfolding the spectra. When the level density is known, unfolding is done by using the integrated level density to transform the eigenvalues into dimensionless variables with unit mean spacing. When it is not known, as in most practical cases, one usually approximates the level staircase function by a polynomial. We here study the effect of unfolding procedure on the spectral fluctuation of two systems for which the level density is known asymptotically. The first is a time-reversal-invariant chaotic system, which is modeled in RMT by a Gaussian Orthogonal Ensemble (GOE). The second is the case of chaotic systems in which m quantum numbers remain almost undistorted in the early stage of the stochastic transition. The Hamiltonian of a system may be represented by a block diagonal matrix with m blocks of the same size, in which each block is a GOE. Unfolding is done once by using the asymptotic level densities for the eigenvalues of the m blocks and once by representing the integrated level density in terms of polynomials of different orders. We find that the spacing distribution of the eigenvalues shows a little sensitivity to the unfolding method. On the other hand, the variance of level number {\Sigma}2(L)is sensitive to the choice of the unfolding function. Unfolding that utilizes low order polynomials enhances {\Sigma}2(L) relative to the theoretical value, while the use of high order polynomial reduces it. The optimal value of the order of the unfolding polynomial depends on the dimension of the corresponding ensemble.Comment: 16 pages, 7 figure

Statistical Analysis of Composite Spectra

We consider nearest neighbor spacing distributions of composite ensembles of levels. These are obtained by combining independently unfolded sequences of levels containing only few levels each. Two problems arise in the spectral analysis of such data. One problem lies in fitting the nearest neighbor spacing distribution to the histogram of level spacings obtained from the data. We show that the method of Bayesian inference is superior to this procedure. The second problem occurs when one unfolds such short sequences. We show that the unfolding procedure generically leads to an overestimate of the chaoticity parameter. This trend is absent in the presence of long-range level correlations. Thus, composite ensembles of levels from a system with long-range spectral stiffness yield reliable information about the chaotic behavior of the system.Comment: 26 pages, 3 figures; v3: changed conclusions, appendix adde