MATHPH In the literature: partially resolved

Poisson spacing statistics for generic integrable systems

In plain words

For systems with regular classical motion, quantum levels should be spaced like uncorrelated random numbers. Even for a flat rectangular drum this is only partly proven.

Precise statement

Laplacian on the flat torus with eigenvalues $m^{2} + \alpha n^{2}$ (m, n integers). Prove that the nearest-neighbour spacing distribution of the unfolded eigenvalues is $\exp(-s)$ for almost every $\alpha$, or for an explicit Diophantine $\alpha$. Answer: a proof.

What would settle it

A proof of Poisson higher correlations or spacing distribution for almost every $\alpha$.

Status in the literature

Pair correlation is Poisson for almost every $\alpha$ (Sarnak, 1997) and for explicit Diophantine $\alpha$ (Eskin, Margulis and Mozes, 2005); higher correlations and the spacing distribution remain open.

See also