Nearest-neighbor-spacing distribution of a system with many degrees of freedom, some regular and some chaotic

We consider a quantum system with a Hamiltonian expressed as a sum of two terms. The first is chaotic, and considered a member of a Gaussian orthogonal ensemble (GOE) of random matrices. The second is integrable, having either equally spaced levels or levels with spacings satisfying a Poisson distribution. The resulting nearest-neighbor-spacing (NNS) distribution of the energy levels of the system is nearly Poissonian in both cases when the analysis involves a large number of levels. if a limited number of levels is considered in each case, deviations from the Poisson distribution are observed. When the regular part of the Hamiltonian is an oscillator with a limited number of phonons, the resulting NNS distribution can be considered as a superposition of independent sequences of levels with GOE statistics when the oscillator energy quantum is larger than the mean spacing of the other term of the Hamiltonian. This distribution has a shape intermediate between the Wigner and the Poisson, and gradually approaches the latter when the number of phonons is increased. This transitional behavior is well reproduced by averaging the level-repulsion function, which may be considered as a justification for a method, recently suggested to calculate the NNS distributions for systems with mixed classical dynamics.