STATISTICS OF ENERGY-LEVELS IN INTEGRABLE QUANTUM-SYSTEMS

We investigate various statistics of energy levels of integrable quantum systems with Hamiltonians H = 1/2 (I - alpha)2 on the unit torus, with alpha a parameter. We find strong numerical evidence, by using up to 10(9) levels, that for typical alpha, with respect to uniform distribution in the unit square, the local empirical statistics of the levels E(n) = 1/2 (n - alpha)2 n is-an-element-of Z2, coverge for large energies to a Poisson limit. The fluctuation of the total number of levels, E(n) < E, scales like E 1/4 and its distribution converges to a non-Gaussian limit. The variance and skewness of this distribution can be computed analytically.