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 lambdaE + sE1/4, where lambda is a constant and s a fluctuating quantity with a non-Gaussian distribution. The probability distribution of s decreases roughly like exp(-s4) when s is large. The number of levels between E and E + z square-root E is equal to lambdaz square-root E + rE1/4 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 --> infinity.