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Higher-order energy level spacing distributions in the transition region between regularity and chaos

We study general energy level spacing distributions of Hamiltonian systems in the transition region between regularity and chaos. The well known Brody distribution, which results from a power-law ansatz for the level-repulsion function, describes the nearest-neighbour spectral spacings. We pursue an analogous ansatz to determine the level spacing distributions of the kth neighbours, which describe level correlations on longer ranges. The new formula is tested by way of example of the numerical spectra of two different classically chaotic Hamiltonian systems, namely the hydrogen atom in a magnetic field and the Henon-Heiles system.