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

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.