Cherednik integrable system: eigenfunctions at generic eigenvalues
Abstract
Symmetric Macdonald polynomials of N variables provide eigenfunctions of the N-body trigonometric Ruijsenaars-Schneider integrable system at particular eigenvalues. In order to construct eigenfunctions with arbitrary eigenvalues, M. Noumi and J. Shiraishi used a recursion in N (branching rule) for the symmetric Macdonald polynomials and analytically continued them. This generated a power series, which is a part of triad (universal solution). In the present paper, we demonstrate that a similar procedure is available for another integrable system, N-body Cherednik integrable system inspired by the DAHA of type A, which has non-symmetric Macdonald polynomials as its polynomial eigenfunctions. However, in this system, the generic eigenfunction is more complicated: it is not just a simple power series as in the Noumi-Shiraishi case, but has an involved structure with N! branches, each of them being a sum over the Weyl chambers of power series of the Noumi-Shiraishi type. As an illustration, we also provide explicit formulas for particular cases.
Versions
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BibTeX
@misc{mironov2026cherednik,
title = {{Cherednik integrable system: eigenfunctions at generic eigenvalues}},
author = {A. Mironov and A. Morozov and A. Popolitov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.09203},
eprint = {2607.09203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09203},
}
RIS
TY - UNPB
TI - Cherednik integrable system: eigenfunctions at generic eigenvalues
AU - A. Mironov
AU - A. Morozov
AU - A. Popolitov
PY - 2026
DA - 2026/08/24
PB - arXiv
DO - 10.48550/arxiv.2607.09203
UR - https://arxiv.org/abs/2607.09203
AB - Symmetric Macdonald polynomials of N variables provide eigenfunctions of the N-body trigonometric Ruijsenaars-Schneider integrable system at particular eigenvalues. In order to construct eigenfunctions with arbitrary eigenvalues, M. Noumi and J. Shiraishi used a recursion in N (branching rule) for the symmetric Macdonald polynomials and analytically continued them. This generated a power series, which is a part of triad (universal solution). In the present paper, we demonstrate that a similar procedure is available for another integrable system, N-body Cherednik integrable system inspired by the DAHA of type A, which has non-symmetric Macdonald polynomials as its polynomial eigenfunctions. However, in this system, the generic eigenfunction is more complicated: it is not just a simple power series as in the Noumi-Shiraishi case, but has an involved structure with N! branches, each of them being a sum over the Weyl chambers of power series of the Noumi-Shiraishi type. As an illustration, we also provide explicit formulas for particular cases.
ER -