Explicit solution of the (quantum) elliptic Calogero-Sutherland model

TL;DR

This paper provides explicit formulas for the eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland model, extending the understanding of its spectral properties and connecting to elliptic deformations of Jack polynomials.

Contribution

It introduces an explicit series solution for the model's eigenfunctions and eigenvalues, applicable to arbitrary particles and couplings, and explores convergence properties.

Findings

Eigenfunctions expressed as infinite series valid for all particle numbers

Eigenvalues derived explicitly for the elliptic Calogero-Sutherland model

Convergence proven in special cases including the two-particle Lamé case

Abstract

We derive explicit formulas for the eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland model as infinite series, to all orders and for arbitrary particle numbers and coupling parameters. The eigenfunctions obtained provide an elliptic deformation of the Jack polynomials. We prove in certain special cases that these series have a finite radius of convergence in the nome $q$ of the elliptic functions, including the two particle (= Lam\'e) case for non-integer coupling parameters.