Energy-dependent correlations in the $S$-matrix of chaotic systems

TL;DR

This paper investigates the energy-dependent correlations of the scattering matrix in chaotic systems, providing a series expansion for multi-element correlations that generalizes existing mathematical functions.

Contribution

It introduces a series expansion for correlations involving multiple matrix elements of the $S$-matrix, extending the mathematical framework of random matrix theory.

Findings

Derived infinite series expressions for multi-element correlations

Generalized Weingarten functions for circular ensembles

Provides a new mathematical approach to chaotic scattering correlations

Abstract

The $M$-dimensional unitary matrix $S(E)$, which describes scattering of waves, is a strongly fluctuating function of the energy for complex systems such as ballistic cavities, whose geometry induces chaotic ray dynamics. Its statistical behaviour can be expressed by means of correlation functions of the kind $\left \langle S_{ij}(E+\epsilon)S^\dag_{pq}(E-\epsilon)\right\rangle$, which have been much studied within the random matrix approach. In this work, we consider correlations involving an arbitrary number of matrix elements and express them as infinite series in $1/M$, whose coefficients are rational functions of $\epsilon$. From a mathematical point of view, this may be seen as a generalization of the Weingarten functions of circular ensembles.