Berry–Tabor conjecture for integrable systems

Prove the Berry–Tabor conjecture by showing that the unfolded nearest-neighbour spacing distribution for quantum systems whose classical dynamics are integrable follows Poisson statistics in the semiclassical regime.

Background

The authors compute spectra for circular and oval billiards, which are classically integrable, and observe Poissonian nearest-neighbour spacing distributions. This aligns with the Berry–Tabor conjecture asserting that integrable systems exhibit Poisson statistics.

Despite extensive numerical evidence across many systems, a general rigorous proof of the Berry–Tabor conjecture remains open, and the paper references it explicitly as a conjectured property.