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

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 (= Lamé) case for non-integer coupling parameters.

v1: 17 pages. The solution is given as series in q but only to low order. v2: 30 pages. Results significantly extended. v3: 35 pages. Paper completely revised: the results of v1 and v2 are extended to all orders

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• Source identities and kernel functions for deformed (quantum) Ruijsenaars models
• Elliptic Ruijsenaars difference operators on bounded partitions
• A new treatment for some periodic Schrödinger operators I: the eigenvalue
• Series Solutions of the Non-Stationary Heun Equation
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• Exact solutions by integrals of the non-stationary elliptic Calogero-Sutherland equation
• Conformal field theory, solitons, and elliptic Calogero--Sutherland models