Spectrum and eigenfunctions of the lattice hyperbolic Ruijsenaars-Schneider system with exponential Morse term

TL;DR

This paper analyzes a lattice hyperbolic Ruijsenaars-Schneider quantum system with an exponential Morse term, diagonalizing it using multivariate polynomials, and confirming its integrability and scattering properties.

Contribution

It introduces a new diagonalization method for the lattice hyperbolic Ruijsenaars-Schneider system using multivariate dual $q$-Hahn polynomials, linking it to Macdonald-Koornwinder polynomials.

Findings

Computed the $n$-particle scattering operator.

Identified the bispectral dual system.

Confirmed quantum integrability in a Hilbert space setting.

Abstract

We place the hyperbolic quantum Ruijsenaars-Schneider system with an exponential Morse term on a lattice and diagonalize the resulting $n$-particle model by means of multivariate continuous dual $q$-Hahn polynomials that arise as a parameter reduction of the Macdonald-Koornwinder polynomials. This allows to compute the $n$-particle scattering operator, to identify the bispectral dual system, and to confirm the quantum integrability in a Hilbert space set-up.