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 + epsilon)S dagger(E - epsilon), averaged over E, and by the statistical properties of the time delay operator, Q(E) = i (h) over barS dagger 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 epsilon, and another as a power series in epsilon 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.

Automated summary

This paper investigates the properties of the M-dimensional scattering matrix and energy correlators in chaotic systems. By analyzing the correlators constructed from traces of powers of the matrix and the statistical properties of the time delay operator, two kinds of expressions for the energy correlators are derived. An explicit formula for Tr(Q(m)) is extracted, which is valid for all m and agrees with predictions from random matrix theory.