The Bohigas–Giannoni–Schmit conjecture on chaotic quantum spectra

Let $E_n$ be the eigenvalues of a quantum Hamiltonian describing a particle in a domain, and suppose the corresponding classical system is sufficiently chaotic, with generic spectral statistics. EGO and EGU denote the orthogonal and unitary Gaussian random-matrix ensembles, respectively.

Bohigas–Giannoni–Schmit conjecture. If the classical problem is a sufficiently chaotic system, then the local statistics of the $E_n$ are generically those of EGO or EGU, according to whether time-reversal invariance is respected.

This conjecture predicts that chaotic quantum systems have universal random-matrix spectral statistics. The source gives numerical evidence from chaotic billiards and notes that the genericity condition is necessary.