The quantum dynamics of the compactified trigonometric Ruijs']jsenaars-Schneider model

We quantize a compactified version of the trigonometric Ruijsenaars-Schneider particle model with a phase space that is symplectomorphic to the complex projective space CPN. The quantum Hamiltonian is realized as a discrete difference operator acting in a finite-dimensional Hilbert space of complex functions with support in a finite uniform lattice over a convex polytope (viz,, a restricted Weyl alcove with walls having a thickness proportional to the coupling parameter). We solve the corresponding finite-dimensional (bispectral) eigenvalue problem in terms of discretized Macdonald polynomials with q (and t) on the unit circle. The normalization of the wave functions is determined using a terminating version of a recent summation formula due to Aomoto, Ito and Macdonald. The resulting eigenfunction transform determines a discrete Fourier-type involution in the Hilbert space of lattice functions. This is in correspondence with Ruijsenaars observation that - at the classical level - the action-angle transformation defines an (anti)symplectic involution of CPN. From the perspective of algebraic combinatorics, our results give rise to a novel system of bilinear summation identities for the Macdonald symmetric functions.