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.

References

On the other hand, for generic integrable classical systems, Berry, and Tabor conjectured that the level spacing distribution follows Poisson statistics.

— Manifestations of chaos in billiards: the role of mixed curvature  (2501.08839 - Das et al., 15 Jan 2025) in Subsection 4B (Level spacing distribution)

In 1977 Berry and Tabor conjectured that a generic integrable quantum system has Poissonian local spectral statistics once its exact symmetries have been removed.

— Poissonian pair correlations for the three-particle Sutherland model  (2609.03112 - Lutsko, 2 Sep 2026) in Introduction

For generic classically integrable systems, the Berry--Tabor conjecture instead suggests that the variance of $N_f(\cdot,\eta)$ grows like $\frac{\eta}{h2}$, which is the average number of energy levels in a window of width $\eta$.

— Smooth spectral statistics of random perturbations of Schrödinger operators near Anosov energy levels  (2610.09869 - Moy, 7 Oct 2026) in Section 1, subsection “Spectral fluctuations”

More specifically, the Berry--Tabor conjecture asserts that the spectral statistics of systems whose classical counterparts are integrable are expected to follow those of a Poisson point process.

— On the asymptotic behavior of the number variance of random points on the unit torus  (2610.11687 - Aistleitner et al., 8 Oct 2026) in Section 1, Introduction and statement of results

The Berry–Tabor conjecture [5] predicts Poissonian local statistics for suitably normalized spectra of generic quantum systems whose classical dynamics are integrable.

— Metric Poissonian Pair Correlation Property Part 1: The Case \(n+\log n\)  (2610.22836 - Tan et al., 16 Oct 2026) in Section 1, Introduction, p. 2