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

TL;DR

This paper develops a semiclassical framework to analyze energy correlations in the scattering matrix and time delay in chaotic systems, providing explicit formulas that align with random matrix theory predictions.

Contribution

It introduces a semiclassical method to derive explicit formulas for energy correlators and time delay moments in chaotic systems with broken time reversal symmetry.

Findings

Derived power series expressions for energy correlators in terms of 1/M and ε

Obtained an explicit formula for Tr(Q^n) valid for all n

Results agree with predictions from random matrix theory

Abstract

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, $Q(E)=-i\hbar S^\dagger dS/dE$. 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 $\epsilon$, and another as a power series in $\epsilon$ whose coefficients are rational functions of $M$. From the latter we extract an explicit formula for $\rm{Tr}(Q^n)$ which is valid for all $n$ and is in agreement with random matrix theory predictions.