The Quantum Dynamics of the Compactified Trigonometric Ruijsenaars–Schneider Model

Abstract:

We quantize a compactified version of the trigonometric Ruijsenaars–Schneider particle model with a phase space that is symplectomorphic to the complex projective space ℂℙ^N. 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 ℂℙ^N. From the perspective of algebraic combinatorics, our results give rise to a novel system of bilinear summation identities for the Macdonald symmetric functions.