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Eigenfunctions of a Discrete Elliptic Integrable Particle Model with Hyperoctahedral Symmetry

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Abstract

We show that for a domain of parameter values subject to a truncation condition, a previously introduced elliptic Ruijsenaars type quantum particle hamiltonian with hyperoctahedral symmetry restricts to a self-adjoint discrete difference operator in a finite-dimensional Hilbert space of functions supported on bounded partitions; the construction of an orthogonal eigenbasis diagonalizing the corresponding discrete quantum model hinges in turn on the spectral theorem for self-adjoint operators in finite dimension. We verify that in the trigonometric limit the eigenfunctions in question recover a previously studied q-Racah type reduction of the Koornwinder–Macdonald polynomials. When the interaction between the particles degenerates to a Pauli repulsion of free fermions, the orthogonal eigenbasis can be expressed in terms of generalized Schur polynomials on the spectrum that are associated with recently found elliptic Racah polynomials.

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References

  1. Atai, F.: Source identities and kernel functions for the deformed Koornwinder-van Diejen models. Commun. Math. Phys. 377, 2191–2216 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  2. Chalykh, O.: Quantum Lax pairs via Dunkl and Cherednik operators. Commun. Math. Phys. 369, 261–316 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  3. Fehér, L., Kluck, T.J.: New compact forms of the trigonometric Ruijsenaars–Schneider system. Nuclear Phys. B 882, 97–127 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  4. Fehér, L., Görbe, T.F.: Trigonometric and elliptic Ruijsenaars–Schneider systems on the complex projective space. Lett. Math. Phys. 106, 1429–1449 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  5. Forrester, P.J.: Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882–1883. Random Matrices Theory Appl. 8(2), 1930001 (2019)

    Article  MathSciNet  Google Scholar 

  6. Görbe, T.F., Hallnäs, M.: Quantization and explicit diagonalization of new compactified trigonometric Ruijsenaars–Schneider systems. J. Integrable Syst. 3(1), 29 (2018)

    Article  MathSciNet  Google Scholar 

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

  8. Komori, Y., Hikami, K.: Quantum integrability of the generalized elliptic Ruijsenaars models. J. Phys. A 30, 4341–4364 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  9. Komori, Y., Hikami, K.: Conserved operators of the generalized elliptic Ruijsenaars models. J. Math. Phys. 39, 6175–6190 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  10. Koornwinder, T.H.: Askey-Wilson polynomials for root systems of type BC, in: Hypergeometric functions on domains of positivity, Jack polynomials, and applications, Contemp. Math. 138, Amer. Math. Soc., 189–204 (1992)

  11. Lawden, D.F.: Elliptic Functions and Applications, Applied Mathematical Sciences, vol. 80. Springer, New York (1989)

    Book  Google Scholar 

  12. Macdonald, I.G.: Schur functions: theme and variations. Séminaire Lotharingien de Combinatoire 28, B28a, 5–39 (1992)

  13. Nakagawa, J., Noumi, M., Shirakawa, M., Yamada, Y.: Tableau representation for Macdonald’s ninth variation of Schur functions. In: Kirillov, A.N., Liskova, N. (eds.) Physics and Combinatorics, pp. 180–195. World Sci. Publ., River Edge (2001)

    Chapter  Google Scholar 

  14. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  15. Prasolov, V.V.: Problems and Theorems in Linear Algebra, Translations of Mathematical Monographs, vol. 134. American Mathematical Society, Providence (1994)

    Book  Google Scholar 

  16. Rains, E.M.: Elliptic double affine Hecke algebras. SIGMA Symmet. Integrab. Geom. Methods Appl. 16, 111 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Ruijsenaars, S.N.M.: Finite-dimensional soliton systems. In: Kupershmidt, B. (ed.) Integrable and Superintegrable Systems, pp. 165–206. World Scientific, Singapore (1990)

    Chapter  Google Scholar 

  18. Ruijsenaars, S.: Action-angle maps and scattering theory for some finite-dimensional integrable systems. III. Sutherland type systems and their duals. Publ. Res. Inst. Math. Sci. 31, 247–353 (1995)

    Article  MathSciNet  Google Scholar 

  19. Ruijsenaars, S.N.M.: Integrable \(BC_N\) analytic difference operators: hidden parameter symmetries and eigenfunctions. In: Shabat, A.B., González-López, A., Mañas, M., Martínez Alonso, L., Rodríguez, M.A. (eds.) New Trends in Integrability and Partial Solvability. NATO Sci. Ser. II Math. Phys. Chem., vol. 132, pp. 217–261. Kluwer Acad. Publ., Dordrecht (2004)

  20. Ruijsenaars, S.N.M.: Hilbert-Schmidt operators vs integrable systems of elliptic Calogero-Moser type. I. The eigenfunction identities. Commun. Math. Phys. 286(2), 629–657 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  21. Ruijsenaars, S.: Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type IV. The relativistic Heun (van Diejen) case. SIGMA Symmet. Integrab. Geom. Methods Appl. 11, 78 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Sarkissian, G.A., Spiridonov, V.P.: Complex hypergeometric functions and integrable many body problems. arXiv: 2105.15031

  23. Sergeev, A.N., Veselov, A.P.: Jacobi–Trudy formula for generalized Schur polynomials. Mosc. Math. J. 14, 161–168 (2014)

    Article  MathSciNet  Google Scholar 

  24. Spiridonov, V.P.: Elliptic hypergeometric functions and models of Calogero–Sutherland type. Teoret. Mat. Fiz. 150, 311–324 (2007)

    Article  MathSciNet  Google Scholar 

  25. Szegö, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society, Providence (1975)

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

    Article  ADS  MathSciNet  Google Scholar 

  27. van Diejen, J.F.: Difference Calogero–Moser systems and finite Toda chains. J. Math. Phys. 36, 1299–1323 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  28. van Diejen, J.F., Görbe, T.: Elliptic Racah polynomials, preprint (2021). arXiv:2106.07394

  29. van Diejen, J.F., Görbe, T.: Elliptic Ruijsenaars difference operators on bounded partitions. Int. Math. Res. Not. IMRN (2021). https://doi.org/10.1093/imrn/rnab251

    Article  Google Scholar 

  30. van Diejen, J.F., Stokman, J.V.: Multivariable \(q\)-Racah polynomials. Duke Math. J. 91, 89–136 (1998)

    MathSciNet  MATH  Google Scholar 

  31. van Diejen, J.F., Vinet, L.: The quantum dynamics of the compactified trigonometric Ruijsenaars–Schneider model. Commun. Math. Phys. 197, 33–74 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  32. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927)

    MATH  Google Scholar 

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Acknowledgements

Thanks are due to an anonymous referee for suggesting some important improvements in the presentation. The work of JFvD was supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grant # 1210015. TG was supported in part by the NKFIH Grant K134946.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile

    Jan Felipe van Diejen

  2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, PO Box 407, 9700, AK, Groningen, The Netherlands

    Tamás Görbe

Authors
  1. Jan Felipe van Diejen
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  2. Tamás Görbe
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Correspondence to Jan Felipe van Diejen.

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Communicated by J. de Gier.

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van Diejen, J.F., Görbe, T. Eigenfunctions of a Discrete Elliptic Integrable Particle Model with Hyperoctahedral Symmetry. Commun. Math. Phys. 392, 279–305 (2022). https://doi.org/10.1007/s00220-022-04350-9

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  • DOI: https://doi.org/10.1007/s00220-022-04350-9

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