Eigenfunctions of a discrete elliptic integrable particle model with hyperoctahedral symmetry

TL;DR

This paper constructs an orthogonal eigenbasis for a discrete elliptic quantum particle model with hyperoctahedral symmetry, connecting it to known polynomials in special limits and degenerations.

Contribution

It introduces a new eigenbasis for a discrete elliptic Ruijsenaars type Hamiltonian with hyperoctahedral symmetry, linking it to Koornwinder-Macdonald and elliptic Racah polynomials.

Findings

Eigenfunctions recover q-Racah type polynomials in the trigonometric limit

Eigenbasis simplifies to generalized Schur polynomials in the impenetrable boson limit

Connects elliptic integrable models with classical orthogonal polynomials

Abstract

We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In the trigonometric limit the eigenfunctions in question recover a previously studied $q$-Racah type reduction of the Koornwinder-Macdonald polynomials. When the inter-particle interaction degenerates to that of impenetrable bosons, the orthogonal eigenbasis simplifies in terms of generalized Schur polynomials on the spectrum associated with recently found elliptic Racah polynomials.