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 XE+sE', where A, is a constant and s a Auctuating quantity with a non-Gaussian distribution. The probability distribution of s decreases roughly like exp( —s ) when s is large. The number of levels between E and E+zJE is equal to kz JE+rE 'I where r is another lluctuating quantity. The distribution of r tends to a Gaussian distribution as z 0 and oscillates around some limiting non-Gaussian distribution as z