On the explicit solutions of the elliptic Calogero system

Abstract

Let $q_1,q_2,...,q_N$ be the coordinates of $N$ particles on the circle, interacting with the integrable potential $\sum_{j<k}^N\wp(q_j-q_k)$, where $\wp$ is the Weierstrass elliptic function. We show that every symmetric elliptic function in $q_1,q_2,...,q_N$ is a meromorphic function in time. We give explicit formulae for these functions in terms of genus $N-1$ theta functions.