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

The $M$-dimensional scattering matrix $S(E)$ which connects incoming to outgoing waves in a chaotic systyem is always unitary, but shows complicated dependence on the energy. This is partly encoded in correlators constructed from traces of powers of $S(E+ε)S^\dagger(E-ε)$, 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 with broken time reversal symmetry, we derive two kind of expressions for the energy correlators: one as a power series in $1/M$ whose coefficients are rational functions of $ε$, and another as a power series in $ε$ whose coefficients are rational functions of $M$. From the latter we extract an explicit formula for $\rm{Tr}(Q^n)$ which is valid for all $n$ and is in agreement with random matrix theory predictions.