Tokai University, Japan

We report our recent studies on the energy level statistics of integrable quantum systems. Based on the assumption proposed by Berry and Robnik, level spacing distribution of a system consisting of infinitely many independent components is derived as a weak limit and its deviations from the Poisson distribution is discussed. The limiting level spacing distribution is specified by a single monotonically increasing function of the level spacing S. Three cases are distinguished by using : (i) Poissonian if , (ii) Poissonian for large S, but possibly not for small S if , and (iii) sub-Poissonian if . This implies that, even when energy level distributions of individual components are statistically independent, the Berry-Robnik approach would provide level spacing distributions other than the Poissonian distribution.