Non-Gaussian energy level statistics for some integrable systems

The number of levels with energy less than E of an integrable quantum system with two degrees of freedom is equal to λE+${\mathit{sE}}^{1/4}$, where λ is a constant and s a fluctuating quantity with a non-Gaussian distribution. The probability distribution of s decreases roughly like exp(-${\mathit{s}}^4$) when s is large. The number of levels between E and E+z √E is equal to λz √E +${\mathit{rE}}^{1/4}$ where r is another fluctuating quantity. The distribution of r tends to a Gaussian distribution as z→0 and oscillates around some limiting non-Gaussian distribution as z→∞.