HIERARCHICAL LEVEL-CLUSTERING IN 2-DIMENSIONAL HARMONIC-OSCILLATORS

We present numerical results for the statistical distribution of energ y level spacings in two-dimensional harmonic oscillators with the irra tional frequency ratio R=omega(1)/omega(2). Unlike scaled level spacin gs, the distribution of the true energy level spacings is well behaved , and directly reflects the corresponding classical quasiperiodic moti on. The histogram of the energy level spacings shows sharp peaks at di scontinuous values which form a hierarchical rational approximations t o R. The peak heights follow a characteristic inverse-square-law incre ase as the level spacing Delta epsilon 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 dist ribution is due to the lack of proper energy scales in the system, sin ce the average (true) level spacing vanishes in the semiclassical limi t.