Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 105, 26 pages      arXiv:2006.07171      https://doi.org/10.3842/SIGMA.2020.105
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

Basic Properties of Non-Stationary Ruijsenaars Functions

Edwin Langmann ^a, Masatoshi Noumi ^b and Junichi Shiraishi ^c
a) Physics Department, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden
^b) Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
^c) Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan

Received June 15, 2020, in final form October 08, 2020; Published online October 21, 2020

Abstract
For any variable number, a non-stationary Ruijsenaars function was recently introduced as a natural generalization of an explicitly known asymptotically free solution of the trigonometric Ruijsenaars model, and it was conjectured that this non-stationary Ruijsenaars function provides an explicit solution of the elliptic Ruijsenaars model. We present alternative series representations of the non-stationary Ruijsenaars functions, and we prove that these series converge. We also introduce novel difference operators called ${\mathcal T}$ which, as we prove in the trigonometric limit and conjecture in the general case, act diagonally on the non-stationary Ruijsenaars functions.

Key words: elliptic integrable systems; elliptic hypergeometric functions; Ruijsenaars systems.

pdf (525 kb)   tex (28 kb)  

References

1. Atai F., Langmann E., Series solutions of the non-stationary Heun equation, SIGMA 14 (2018), 011, 32 pages, arXiv:1609.02525.
2. Atai F., Langmann E., Exact solutions by integrals of the non-stationary elliptic Calogero-Sutherland equation, J. Integrable Syst. 5 (2020), xyaa001, 26 pages, arXiv:1908.00529.
3. Awata H., Kanno H., Mironov A., Morozov A., On a complete solution of the quantum Dell system, J. High Energy Phys. 2020 (2020), no. 4, 212, 30 pages, arXiv:1912.12897.
4. Braden H.W., Marshakov A., Mironov A., Morozov A., On double-elliptic integrable systems. I. A duality argument for the case of ${\rm SU}(2)$, Nuclear Phys. B 573 (2000), 553-572, arXiv:hep-th/9906240.
5. Felder G., Varchenko A., The $q$-deformed Knizhnik-Zamolodchikov-Bernard heat equation, Comm. Math. Phys. 221 (2001), 549-571, arXiv:math.QA/9809139.
6. Felder G., Varchenko A., $q$-deformed KZB heat equation: completeness, modular properties and ${\rm SL}(3,{\mathbb Z})$, Adv. Math. 171 (2002), 228-275, arXiv:math.QA/0110081.
7. Felder G., Varchenko A., Hypergeometric theta functions and elliptic Macdonald polynomials, Int. Math. Res. Not. 2004 (2004), 1037-1055, arXiv:math.QA/0309452.
8. Fock V., Gorsky A., Nekrasov N., Rubtsov V., Duality in integrable systems and gauge theories, J. High Energy Phys. 2000 (2000), no. 7, 028, 40 pages, arXiv:hep-th/9906235.
9. Fukuda M., Ohkubo Y., Shiraishi J., Non-stationary Ruijsenaars functions for $\kappa=t^{-1/N}$ and intertwining operators of Ding-Iohara-Miki algebra, arXiv:2002.00243.
10. Hallnäs M., Ruijsenaars S., Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics, arXiv:1905.12918.
11. Koroteev P., Shakirov S., The quantum DELL system, Lett. Math. Phys. 110 (2020), 969-999, arXiv:1906.10354.
12. Langmann E., Explicit solution of the (quantum) elliptic Calogero-Sutherland model, Ann. Henri Poincaré 15 (2014), 755-791, arXiv:math-ph/0401029.
13. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
14. Nekrasov N.A., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), 831-864, arXiv:hep-th/0206161.
15. Noumi M., Shiraishi J., A direct approach to the bispectral problem for the Ruijsenaars-Macdonald $q$-difference operators, arXiv:1206.5364.
16. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
17. Ruijsenaars S.N.M., Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987), 191-213.
18. Ruijsenaars S.N.M., Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type. I. The eigenfunction identities, Comm. Math. Phys. 286 (2009), 629-657.
19. Ruijsenaars S.N.M., Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type. II. The $A_{N-1}$ case: first steps, Comm. Math. Phys. 286 (2009), 659-680.
20. Shiraishi J., A conjecture about raising operators for Macdonald polynomials, Lett. Math. Phys. 73 (2005), 71-81, arXiv:math.QA/0503727.
21. Shiraishi J., Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions, J. Integrable Syst. 4 (2019), xyz010, 30 pages, arXiv:1903.07495.