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.
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