E(K, L) level statistics of classically integrable quantum systems based on the Berry–Robnik approach

The theory of the quantal level statistics of a classically integrable system, developed by Makino et al. in order to investigate the non-Poissonian behaviors of level-spacing distribution (LSD) and level-number variance (LNV) [H. Makino and S. Tasaki, Phys. Rev. E 67, 066205 (2003); H. Makino and S. Tasaki, Prog. Theor. Phys. Suppl. 150, 376 (2003); H. Makino, N. Minami, and S. Tasaki, Phys. Rev. E 79, 036201 (2009); H. Makino and S. Tasaki, Prog. Theor. Phys. 114, 929 (2005)], is successfully extended to the study of the $E(K,L)$ function, which constitutes a fundamental measure to determine most statistical observables of quantal levels in addition to LSD and LNV. In the theory of Makino et al., the eigenenergy level is regarded as a superposition of infinitely many components whose formation is supported by the Berry–Robnik approach in the far semiclassical limit [M. Robnik, Nonlinear Phenom. Complex Syst. 1, 1 (1998)]. We derive the limiting $E(K,L)$ function in the limit of infinitely many components and elucidate its properties when energy levels show deviations from the Poisson statistics.