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

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.

💡 Research Summary

The paper investigates a lattice version of the hyperbolic quantum Ruijsenaars‑Schneider (RS) system augmented by an exponential Morse potential. Starting from the continuous hyperbolic RS model, the authors introduce a discrete lattice by replacing the differential operators with finite‑difference operators and by defining a Hamiltonian that incorporates both the hyperbolic interaction (parameterized by q and t) and the Morse term (controlled by four coupling parameters t₀,…,t₃). After fixing one of the Morse parameters to unity, the model is expressed in a compact form that is amenable to exact analysis.

The central technical achievement is the diagonalization of this lattice Hamiltonian using multivariate continuous dual q‑Hahn polynomials. These polynomials arise as a parameter reduction of the Macdonald‑Koornwinder (multivariate Askey‑Wilson) polynomials. They are orthogonal with respect to a weight function ˆΔ(ξ) defined on the alcove A = {π > ξ₁ > … > ξₙ > 0}. By normalizing the leading coefficient so that P_λ(i ρ̂)=1, the authors construct a Fourier kernel ψ_ξ(ρ+λ)=P_λ(ξ) which implements a unitary map between the Hilbert space ℓ²(ρ+Λ,Δ) of lattice functions and L²(A,ˆΔ).

Theorem 1 shows that the lattice Hamiltonian H is unitarily equivalent to a bounded multiplication operator ˆE(ξ) = Σ_{j=1}ⁿ