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Construction of Eigenfunctions for the Elliptic Ruijsenaars Difference Operators

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Abstract

We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic deformations of the Macdonald polynomials, and the second kind generalizes asymptotically free eigenfunctions previously constructed in the trigonometric case. We obtain these eigenfunctions as infinite series which, as we show, converge in suitable domains of the variables and parameters. Our results imply that, for the domain where the elliptic Ruijsenaars operators define a relativistic quantum mechanical system, the elliptic deformations of the Macdonald polynomials provide a family of orthogonal functions with respect to the pertinent scalar product.

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Notes

  1. We mention in passing that m, c and g have the physical interpretation of particle mass, vacuum velocity of light, and coupling constant, respectively [18]; note that \(u_i\) here corresponds to \(q_i\) in [18].

  2. This relation between the trigonometric Ruijsenaars system and the Macdonald polynomials was already noted in by Koornwinder in 1987 [25].

  3. The precise relation is \(P_\lambda (x|q,t) = x^\lambda f(x;t^{\rho }q^{\lambda }|q,t)\) with \(\rho _i = n-i\), using a common shorthand notation introduced in the main text; see [17] for the proof of this fact.

  4. Note that f(x; s|q, t) here is denoted as \(p_n(x; s|q, t)\) in [17].

  5. We hope that no confusion would arise with the notation \(q^{S}\) introduced in (1).

  6. Any joint eigenfunction of the operators \({\mathcal {D}}^{(r)}_x(p)\) (\(r=1,\ldots ,n\)) is necessarily an eigenfunction of \({\mathcal {D}}^{(r)}_x(p)\) for \(r=-1,\ldots ,-n\) as well. For this reason, we will mainly consider the elliptic Ruijsenaars operators \({\mathcal {D}}^{(r)}_x(p)\) (\(r=1,\ldots ,n\)) of positive orders in the construction of joint eigenfunctions.

  7. Here, the subcript 0 stands for the empty partition.

  8. The left-hand side in (5.101) would perhaps be better written as \(\left\langle \varphi (\cdot ;p),\psi (\cdot ;p)\right\rangle \), but we find it convenient to abuse the notation here.

  9. The functions \(f^{\widehat{\mathfrak {gl}}_n}\) and \(f_{n,\infty }\) are related to each other by a simple change of variables [13, Theorem 3.2].

  10. We are grateful to S. Ruijsenaars for emphasizing to us the importance of this symmetry.

  11. Our current proof of this is lengthy, and we therefore do not include it in the present paper.

References

  1. Cherednik, I.: Whittaker limits of difference spherical functions. Int. Math. Res. Not. IMRN, pp. 3793–3842 (2009)

  2. Felder, G., Varchenko, A.: Elliptic quantum groups and Ruijsenaars models. J. Statist. Phys. 89, 963–980 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  3. Felder, G., Varchenko, A.: Algebraic integrability of the two-particle Ruijsenaars operator. Funct. Anal. Appl. 32, 81–92 (1998)

    Article  MathSciNet  Google Scholar 

  4. Felder, G., Varchenko, A.: Hypergeometric theta functions and elliptic Macdonald polynomials, Int. Math. Res. Not. IMRN (2004), pp. 1037–1055

  5. Hasegawa, K.: Ruijsenaars’ commuting difference operators as commuting transfer matrices. Commun. Math. Phys. 187, 289–325 (1997)

  6. Hallnäs, M., Ruijsenaars, S.: Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics. Int. Math. Res. Not. IMRN (2020). https://doi.org/10.1093/imrn/rnaa193, arXiv:1905.12918

  7. Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd edn, xxii+400 pp. Cambridge University Press, Cambridge (1990)

  8. Kato, T.: Perturbation Theory for Linear Operators. Reprint of the 1980 edition, Classics in Mathematics. Springer, Berlin (1995)

  9. Komori, Y., Noumi, M., Shiraishi, J.: Kernel functions for difference operators of Ruijsenaars type and their applications. SIGMA 5, 054, 40 pages (2009) arXiv:0812.0279

  10. Komori, Y., Takemura, K.: The perturbation of the quantum Calogero–Moser–Sutherland system and related results. Commun. Math. Phys. 227, 93–118 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  11. Langmann, E.: Anyons and the elliptic Calogero-Sutherland model. Lett. Math. Phys. 54, 279–289 (2000)

    Article  MathSciNet  Google Scholar 

  12. Langmann, E.: Explicit solution of the (quantum) elliptic Calogero-Sutherland model. Ann. Henri Poincaré 15, 755–791 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  13. Langmann, E., Noumi, M., Shiraishi, J.: Basic properties of non-stationary Ruijsenaars functions. SIGMA 16, 105, 26 pages (2020). arXiv:2006.07171

  14. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Mathematical Monographs, x+475pp. Oxford University Press, Oxford (1995)

  15. Macdonald, I.G.: Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, vol. 157, pp. x+175. Cambridge University Press, Cambridge (2003)

  16. Mimachi, K., Noumi, M.: An integral representation of eigenfunctions for Macdonald’s \(q\)-difference operators. Tôhoku Math. J. 49, 517–525 (1997)

  17. Noumi, M., Shiraishi, J.: A direct approach to the bispectral problem for the Ruijsenaars-Macdonald \(q\)-difference operators. arXiv:1206.5364, 44 pages

  18. Ruijsenaars, S.N.M.: Complete integrability of relativistic Calogero-Moser systems and elliptic function identities. Commun. Math. Phys. 110, 191–213 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  19. Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  20. Ruijsenaars, S.N.M.: Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero–Moser type, I. The eigenfunction identities. Commun. Math. Phys. 286, 629–657 (2009)

  21. Ruijsenaars, S.N.M.: Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type, II. The \(A_{N-1}\) case: first steps. Commun. Math. Phys. 286, 659–680 (2009)

  22. Shiraishi, J.: A conjecture about raising operators for Macdonald polynomials. Lett. Math. Phys. 73, 71–81 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  23. Shiraishi, J.: Affine screening operators, affine Laumon spaces, and conjectures concerning non-stationary Ruijsenaars functions. J. Integrable Syst. 4, xyz010, 30pp (2019). arXiv:1903.07495, 26 pages

  24. van Diejen, J.F.: Integrability of difference Calogero-Moser systems. J. Math. Phys. 35, 2983–3004 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  25. van Diejen, J.F.: Commuting difference operators with polynomial eigenfunctions. Compositio Math. 95, 183–233 (1995)

    MathSciNet  MATH  Google Scholar 

  26. van Meer, M., Stokman, J.: Double affine Hecke algebras and bispectral quantum Knizhnik-Zamolodchikov equations. Int. Math. Res. Not. IMRN 969–1040 (2010)

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Acknowledgements

This work is supported by VR Grant No. 2016-05167 (E.L.) and by JSPS Kakenhi Grants (B) 15H03626 (M.N.), (C) 19K03512 (J.S.). M.N. is grateful to the Knut and Alice Wallenberg Foundation for funding his guest professorship at KTH. We are grateful to the Stiftelse Olle Engkist Byggmästare, Contract 184-0573, for a travel grant that initiated our collaboration.

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Authors and Affiliations

  1. Department of Physics, KTH Royal Institute of Technology, 106 91, Stockholm, Sweden

    Edwin Langmann

  2. Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden

    Masatoshi Noumi

  3. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Junichi Shiraishi

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  1. Edwin Langmann
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  2. Masatoshi Noumi
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Correspondence to Masatoshi Noumi.

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Communicated by J. d-Gier.

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Masatoshi Noumi: On leave from: Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan.

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Langmann, E., Noumi, M. & Shiraishi, J. Construction of Eigenfunctions for the Elliptic Ruijsenaars Difference Operators. Commun. Math. Phys. 391, 901–950 (2022). https://doi.org/10.1007/s00220-021-04195-8

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