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The $M$-dimensional scattering matrix $S(E)$ which connects incoming tooutgoing waves in a chaotic systyem is always unitary, but shows complicateddependence on the energy. This is partly encoded in correlators constructedfrom traces of powers of $S(E+\epsilon)S^\dagger(E-\epsilon)$, averaged over$E$, and by the statistical properties of the time delay operator,$Q(E)=-i\hbar S^\dagger dS/dE$. Using a semiclassical approach for systems withbroken time reversal symmetry, we derive two kind of expressions for the energycorrelators: one as a power series in $1/M$ whose coefficients are rationalfunctions of $\epsilon$, and another as a power series in $\epsilon$ whosecoefficients are rational functions of $M$. From the latter we extract anexplicit formula for $\rm{Tr}(Q^n)$ which is valid for all $n$ and is inagreement with random matrix theory predictions.

Semiclassical approach to $S$ matrix energy correlations and time delay in chaotic systems