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-α${)}^2$ on the unit torus, with α a parameter. We find strong numerical evidence, by using up to ${10}^9$ levels, that for typical α, with respect to uniform distribution in the unit square, the local empirical statistics of the levels ${\mathit{E}}_{\mathbf{n}}$=1/2(n-α${)}^2$, n∈${\mathit{openZ}}^2$, converge for large energies to a Poisson limit. The fluctuation of the total number of levels, ${\mathit{E}}_{\mathit{n}}$E, scales like ${\mathit{E}}^{1/4}$ and its distribution converges to a non-Gaussian limit. The variance and skewness of this distribution can be computed analytically.