The Bohigas–Giannoni–Schmit (BGS) conjecture also known as the random matrix conjecture) for simple quantum mechanical systems (ergodic with a classical limit) few degrees of freedom holds that spectra of time reversal-invariant systems whose classical analogues are K-systems show the same fluctuation properties as predicted by the GOE (Gaussian orthogonal ensembles).
Alternatively, the spectral fluctuation measures of a classically chaotic quantum system coincide with those of the canonical random-matrix ensemble in the same symmetry class (unitary, orthogonal, or symplectic).
That is, the Hamiltonian of a microscopic analogue of a classical chaotic system can be modeled by a random matrix from a Gaussian ensemble as the distance of a few spacings between eigenvalues of a chaotic Hamiltonian operator generically statistically correlates with the spacing laws for eigenvalues of large random matrices.
A simple example of the unfolded quantum energy levels in a classically chaotic system correlating like that would be Sinai billiards:
Energy levels: Spectral density: Average spectral density: Correlation: Unfolding: Unfolded correlation: BGS conjecture:
The conjecture remains unproven despite supporting numerical evidence.
References
Links the BGS conjecture in Scholarpedia.