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The eigenvalues of quantum chaotic systems have been conjectured to follow,in the large energy limit, the statistical distribution of eigenvalues ofrandom ensembles of matrices of size $N\rightarrow\infty$. Here we providesemiclassical arguments that extend the validity of this correspondence tofinite energies. We conjecture that the spectrum of a generic fully chaoticsystem without time-reversal symmetry has, around some large but finite energy$E$, the same statistical properties as the Circular Unitary Ensemble of randommatrices of dimension $N_{\rm eff} = \tH / \sqrt{24 d_1}$, where $\tH$ isHeisenberg time and $\sqrt{d_1}$ is a characteristic classical time, bothevaluated at energy $E$. A corresponding conjecture is also made for chaoticmaps.

Generalized random matrix conjecture for chaotic systems