Chaos and Regularity in the Doubly Magic Nucleus ^{208}Pb

High-resolution experiments have recently lead to a complete identification (energy, spin, and parity) of 151 nuclear levels up to an excitation energy of E_{x}=6.20  MeV in ^{208}Pb [Heusler et al., Phys. Rev. C 93, 054321 (2016)PRVCAN2469-998510.1103/PhysRevC.93.054321]. We present a thorough study of the fluctuation properties in the energy spectra of the unprecedented set of nuclear bound states. In a first approach, we group states with the same spin and parity into 14 subspectra, analyze standard statistical measures for short- and long-range correlations, i.e., the nearest-neighbor spacing distribution, the number variance Σ^{2}, the Dyson-Mehta Δ_{3} statistics, and the novel distribution of the ratios of consecutive spacings of adjacent energy levels in each energy sequence, and then compute their ensemble average. Their comparison with a random matrix ensemble which interpolates between Poisson statistics expected for regular systems and the Gaussian orthogonal ensemble (GOE) predicted for chaotic systems shows that the data are well described by the GOE. In a second approach, following an idea of Rosenzweig and Porter [Phys. Rev. 120, 1698 (1960)PHRVAO0031-899X10.1103/PhysRev.120.1698], we consider the complete spectrum composed of the independent subspectra. We analyze their fluctuation properties using the method of Bayesian inference involving a quantitative measure, called the chaoticity parameter f, which also interpolates between Poisson (f=0) and GOE statistics (f=1). It turns out to be f≈0.9. This is so far the closest agreement with a GOE observed in the spectra of bound states in a nucleus. The same analysis is also performed with spectra computed on the basis of shell model calculations with different interactions (surface-delta interaction, Kuo-Brown, Michigan-three-Yukawa). While the simple surface-delta interaction exhibits features typical for nuclear many-body systems with regular dynamics, the other, more realistic interactions yield chaoticity parameters f close to the experimental values.