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

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.