Level statistics for two-dimensional oscillators

We consider the level statistics of two-dimensional harmonic oscillators with incommensurable frequencies, which are known to have picket-fence type spectra. We propose a parametric representation for the level-spacing distribution and level-number variance, and study the variation of the parameters with the frequency ratio and the size of the spectra. By introducing an anharmonic perturbation, we observe a gradual transition to the Poisson statistics. We describe the level spectra in transition from harmonic to Poissonian statistics as a superposition of two independent sequences, one for each of the two extreme statistics. We show that this transition provides a suitable description for the evolution of the spectrum of a disordered chain with increasing long range correlations between the lattice sites.