Spectral statistics of the bootstrapped density of states in the d-dimensional integrable billiard system

The bootstrapped density of states introduced by Bogomolny and Keating for classically chaotic systems is applied to integrable systems, and the condition which leads to the universal nature of correlation between levels is investigated. We consider the two-point correlation function of the d-dimensional billiard system. We find the condition between the dimension and the periodic orbit sum in the semiclassical trace formula under which condition the correlation is well described by the Poisson statistics.