Construction of eigenfunctions for the elliptic Ruijsenaars difference operators

TL;DR

This paper develops a perturbative method to construct eigenfunctions of elliptic Ruijsenaars difference operators, extending Macdonald polynomials and their asymptotic generalizations, with convergence and orthogonality properties.

Contribution

It introduces a new perturbative approach to construct eigenfunctions for elliptic Ruijsenaars operators, including elliptic deformations of Macdonald polynomials.

Findings

Eigenfunctions expressed as convergent infinite series.

Elliptic deformations of Macdonald polynomials are orthogonal functions.

Generalization of asymptotically free eigenfunctions to the elliptic case.

Abstract

We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic deformations of the Macdonald polynomials, and the second kind generalizes asymptotically free eigenfunctions previously constructed in the trigonometric case. We obtain these eigenfunctions as infinite series which, as we show, converge in suitable domains of the variables and parameters. Our results imply that, for the domain where the elliptic Ruijsenaars operators define a relativistic quantum mechanical system, the elliptic deformations of the Macdonald polynomials provide a family of orthogonal functions with respect to the pertinent scalar product.