Limiting level spacing distribution of integrable quantum systems

Hironori Makino

Department of Human and Information Science
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 $m(S)$ of the level spacing S. Three cases are distinguished by using $m = m(\infty)$ : (i) Poissonian if $m = 0$, (ii) Poissonian for large S, but possibly not for small S if $0 < m(\infty) < 1$, and (iii) sub-Poissonian if $m(\infty) = 1$. 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.