Hierarchical level-clustering in two-dimensional harmonic oscillators

Abstract

We present numerical results for the statistical distribution of energy level spacings in two-dimensional harmonic oscillators with the irrational frequency ratio R\ensuremath{\equiv}${\mathrm{\ensuremath{\omega}}}_{1}$/${\mathrm{\ensuremath{\omega}}}_{2}$. Unlike scaled level spacings, the distribution of the true energy level spacings is well behaved, and directly reflects the corresponding classical quasiperiodic motion. The histogram of the energy level spacings shows sharp peaks at discontinuous values which form a hierarchical rational approximations to R. The peak heights follow a characteristic inverse-square-law increase as the level spacing \ensuremath{\Delta}E decreases, indicating a form of level clustering rather than level repulsion as previously believed. We believe the failure of convergence in the scaled level spacing distribution is due to the lack of proper energy scales in the system, since the average (true) level spacing vanishes in the semiclassical limit.