On the Two Gap Locus for the Elliptic Calogero--Moser Model

TL;DR

This paper provides an analytical description of the two-gap elliptic potentials in the elliptic Calogero--Moser system, including explicit formulas and a new Lax representation, extending to more general potentials via theta function reductions.

Contribution

It introduces explicit formulas for covers, wave functions, and Lamé polynomials, along with a new Lax representation, and explores reductions to lower genera in the moduli space.

Findings

Explicit formulas for covers, wave functions, and Lamé polynomials.

A new Lax representation for particle dynamics.

Reduction conditions in the moduli space of genus two curves.

Abstract

We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero--Moser system. We start with the description of Treibich--Verdier two--gap elliptic potentials. The explicit formulae for the covers, wave functions and Lam\'e polynomials are derived, together with a new Lax representation for the particle dynamics on the locus. Then we consider more general potentials within the Weierstrass reduction theory of theta functions to lower genera. The reduction conditions in the moduli space of the genus two algebraic curves are given. This is some subvariety of the Humbert surface, which can be singled out by the condition of the vanishing of some theta constants.