Level spacings for harmonic-oscillator systems

Abstract

From the viewpoint of eigenvalue level statistics, harmonic-oscillator systems are unusual. Although integrable, these systems are nongeneric, and a spacing distribution does not exist even as the number of levels N\ensuremath{\rightarrow}\ensuremath{\infty}. The origins of this pathological behavior are explored using methods of number theory and ergodic analysis. However, such nongenericity is extremely fragile, and the smallest nonlinearity asymptotically restores generic behavior. These results are of relevance to the study of molecular spectra, as well as to the quasienergy spectra of integrable quantum maps.