The Calogero-Moser equation system and the ensemble average in the Gaussian ensembles

TL;DR

This paper links the Calogero-Moser equations to Gaussian ensembles in random matrix theory, providing solutions and an expansion for matrix averages using eigenfunctions, thus simplifying the analysis of ensemble averages.

Contribution

It generalizes the Calogero-Moser equations for all coupling constants and applies them to compute ensemble averages in Gaussian ensembles.

Findings

Solutions to CM equations via Rodriguez relation for all coupling constants

Derived an expansion for matrix averages in Gaussian ensembles

Simplified analysis of ensemble averages using eigenfunctions

Abstract

From random matrix theory it is known that for special values of the coupling constant the Calogero-Moser (CM) equation system is nothing but the radial part of a generalized harmonic oscillator Schroedinger equation. This allows an immediate construction of the solutions by means of a Rodriguez relation. The results are easily generalized to arbitrary values of the coupling constant. By this the CM equations become nearly trivial. As an application an expansion for <exp[i(XY)]> in terms of eigenfunctions of the CM equation system is obtained, where X and Y are matrices taken from one of the Gaussian ensembles, and the brackets denote an average over the angular variables.