Level correlation in coupled harmonic oscillator systems

From the viewpoint of nearest neighbour energy levels statistics, harmonic oscillator systems are unusual. Although it is integrable but does not follow the generic rule and the precise form of p(s) distribution depends on the arithmetic nature of the irrational frequency ratios. So from the old studies in this direction no deterministic conclusions have been gained so far and it needs further study in some new direction for its complete resolution. Spectral rigidity is often used as a stronger tool than the p(s) distribution as it can easily take account of long range correlations. Here we present correlation between levels in both small and large energy scale for coupled and uncoupled 2D HO. Instead of true level spacings we use scaled level spacings and our numerical results nicely show that in HO systems energy levels are highly correlated, regular and very rigid.