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

Journal article
Authors Tamas Görbe
Martin Hallnäs
Published in Journal of Integrable Systems
Volume 3
Issue 1
Publication year 2018
Published at Department of Mathematical Sciences
Language en
Links https://doi.org/10.1093/integr/xyy0...
Keywords quantization, Macdonald polynomials, Ruijsenaars–Schneider, Calogero–Moser–Sutherland
Subject categories Mathematical Analysis, Other Mathematics

Abstract

Recently, Fehér 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 An−1 Macdonald polynomials with unitary parameters.

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