Eigenfunctions of the van Diejen model generated by gauge and integral transformations

TL;DR

This paper constructs explicit eigenfunctions for the van Diejen $BC_m$ relativistic Calogero-Moser-Sutherland model using gauge and integral transformations, revealing connections to elliptic hypergeometric integrals and the $E_8$ Weyl group.

Contribution

It introduces a method to generate eigenfunctions of the van Diejen model via gauge and integral transformations, linking them to elliptic hypergeometric integrals and Weyl group symmetries.

Findings

Eigenfunctions expressed as elliptic hypergeometric integrals.

Connection between eigenfunctions and $E_8$ Weyl group reflections.

Identification of joint eigenfunctions for van Diejen operators.

Abstract

We present how explicit eigenfunctions of the principal Hamiltonian for the $BC_{m}$ relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral transformations. In particular, we find that certain $BC$-type elliptic hypergeometric integrals, including elliptic Selberg integrals, of both Type I and Type II arise as eigenfunctions of the van Diejen model under some parameter restrictions. Among these are also joint eigenfunctions of so-called modular pairs of van Diejen operators. Furthermore, these transformations are related to reflections of the $E_{8}$ Weyl group acting on the space of model parameters.