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^{†}(E-ε), averaged over E, and by the statistical properties of the time delay operator, Q(E)=-iℏS^{†}dS/dE. Using a semiclassical approach for systems with broken time-reversal symmetry, we derive two kinds 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 Tr(Q^{m}) which is valid for all m and is in agreement with random matrix theory predictions.