Bohigas–Giannoni–Schmit conjecture for quantum spectra of classically chaotic systems

Prove the Bohigas–Giannoni–Schmit conjecture by establishing that, in the semiclassical limit, the unfolded spectral statistics (including the nearest-neighbour spacing distribution) of quantum Hamiltonians whose classical dynamics are fully chaotic coincide with those of Gaussian random matrix ensembles, with the appropriate symmetry class determined by the presence or absence of time-reversal invariance.

Background

The paper analyses spectral statistics of two mixed-curvature billiards (bean-shaped and peanut-shaped) and finds nearest-neighbour spacing distributions consistent with the Gaussian Orthogonal Ensemble, matching predictions of Random Matrix Theory for time-reversal-invariant chaotic systems. This empirical agreement is discussed in the context of the Bohigas–Giannoni–Schmit (BGS) conjecture, which posits universal random-matrix statistics for quantum systems with classically chaotic dynamics.

While the authors’ numerical results support the conjecture for their specific billiards, the BGS statement remains conjectural in full generality; proving it rigorously for broad classes of chaotic quantum systems is an outstanding problem in quantum chaos.