Quantization and explicit diagonalization of new compactified trigonometric Ruijsenaars-Schneider systems

TL;DR

This paper quantizes new classical compactified trigonometric Ruijsenaars-Schneider systems, explicitly solving their quantum eigenvalue problem using discretized Macdonald polynomials, thus advancing understanding of their spectral properties.

Contribution

It introduces a quantization of type (i) Ruijsenaars-Schneider systems and explicitly constructs their joint eigenfunctions using discretized Macdonald polynomials.

Findings

Quantum Hamiltonians realized as discrete difference operators

Explicit joint eigenfunctions constructed from discretized Macdonald polynomials

Spectral problem solved for these quantum integrable systems

Abstract

Recently, Feh\'er and Kluck discovered, at the level of classical mechanics, new compactified trigonometric Ruijsenaars-Schneider $n$-particle systems, with phase space symplectomorphic to the $(n-1)$-dimensional complex projective space. In this article, we quantize the so-called type (i) instances of these systems and explicitly solve the joint eigenvalue problem for the corresponding quantum Hamiltonians by generalising previous results of van Diejen and Vinet. Specifically, the quantum Hamiltonians are realized as discrete difference operators acting in a finite-dimensional Hilbert space of complex-valued functions supported on a uniform lattice over the classical configuration space, and their joint eigenfunctions are constructed in terms of discretized $A_{n-1}$ Macdonald polynomials with unitary parameters.