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The spectral problem for theq-Knizhnik-Zamolodchikov equation and continuousq-Jacobi polynomials

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Abstract

The spectral problem for theq-Knizhnik-Zamolodchikov equations for\(Uq(\widehat{sl_2 })(0< q< 1)\) at arbitrary non-negative levelk is considered. The case of two-point functions in the fundamental representation is studied in detail. The scattering states are given explicitly in terms of continuousq-Jacobi polynomials, and theS-matrix is derived from their asymptotic behavior. The level zeroS-matrix is closely connected with the kink-antikinkS-matrix for the spin-\(\tfrac{1}{2}\) XXZ antiferromagnet. An interpretation of the latter in terms of scattering on (quantum) symmetric spaces is discussed. In the limit of infinite level we observe connections with harmonic analysis onp-adic groups with the primep given byp=q −2.

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References

  1. Freund, P.G.O., Zabrodin, A.V.: Phys. Lett.B294, 347 (1992)

    Article  Google Scholar 

  2. Freund, P.G.O., Zabrodin, A.V.: Phys. Lett.B311, 103 (1993)

    Article  Google Scholar 

  3. Frenkel, I., Reshetikhin, N.Yu.: Commun. Math. Phys.146, 1 (1992)

    Google Scholar 

  4. Knizhnik, V.G., Zamolodchikov, A.B.: Nucl. Phys.B247, 83 (1984)

    Article  Google Scholar 

  5. Bargmann, V., Wigner, E.P.: Proc. Nat. Acad. Sci. USA34, 211 (1946)

    Google Scholar 

  6. Faddeev, L.D., Takhtajan, A.L.: Zapiski Nauchn. Sem. LOMI109, 134 (1981)

    Google Scholar 

  7. Kirillov, A.N., Reshetikhin, N.Yu.: J. PhysicsA20, 1565 (1987)

    Google Scholar 

  8. Macdonald, I.G.: In: Nevai, P. (ed.) Orthogonal Polynomials: Theory and Practice. Dordrecht: Kluwer Academic, 1990; Macdonald, I.G.: Queen Mary College preprint 1989

    Google Scholar 

  9. Freund, P.G.O.: In: Clavelli, L., Harms, B. (eds.) Superstrings and Particle Theory. Singapore: World Scientific, 1990

    Google Scholar 

  10. Zabrodin, A.V.: Mod. Phys. Lett.A7, 441 (1992)

    Article  Google Scholar 

  11. Freund, P.G.O., Zabrodin, A.V.: Commun. Math. Phys.147, 277 (1992)

    Article  Google Scholar 

  12. Matsuo, A.: Invent. Math.110, 95 (1992)

    Article  Google Scholar 

  13. Cherednik, I.: Preprint RIMS-776 (1991); Duke Math. J.68, 171 (1992)

  14. Erdélyi, A.: Higher Transcendental Functions, v. II. New York: McGraw-Hill, 1953

    Google Scholar 

  15. Davies, B., Foda, O., Jimbo, M., Miwa, T., Nakayashiki, A.: Commun. Math. Phys.151, 89 (1993); Idzumi, M., Iohara, K., Jimbo, M., Miwa, T., Nakashima, T., Tokihiro, T.: RIMS preprint 1992

    Google Scholar 

  16. Rogers, L.J.: Proc. Lond. Math. Soc.26, 318 (1895)

    Google Scholar 

  17. Askey, R., Ismail, M.E.H. In: Erdös, P. (ed.) Studies in Pure Math. Basel: Birkhäuser, 1983

    Google Scholar 

  18. Gasper, G., Rahman, M.: Basic hypergeometric series. Cambridge: Cambridge University Press, 1990

    Google Scholar 

  19. Cherednik, I.: Commun. Math. Phys.150, 109 (1992)

    Google Scholar 

  20. Noumi, M., Mimachi, K.: Proc. Japan Acad. Ser. A66, 146 (1990)

    Google Scholar 

  21. Freund, P.G.O., Zabrodin, A.V.: Phys. Lett.B284, 283 (1992)

    Article  Google Scholar 

  22. Koornwinder, T.: J. Math. Anal. and Appl.148, 44 (1990)

    Article  Google Scholar 

  23. Mautner, F.: Am. J. Math.80, 441 (1958); Cartier, P.: Proc. Symp. Pure Math., vol26, Providence, RI: A.M.S., 1973

    Google Scholar 

  24. Bruhat, F., Tits, J.: Publ. Math. IHES41, 5 (1972)

    Google Scholar 

  25. Matsuo, A.: Commun. Math. Phys.151, 263 (1993)

    Google Scholar 

  26. Rahman, M.: SIAM J. Math. Anal.17, 1280 (1986)

    Article  Google Scholar 

  27. Rahman, M.: Canad. J. Math.33, 255 (1981)

    Google Scholar 

  28. Askey, R., Wilson, J.A.: Memoirs Amer. Math. Soc.319 (1985)

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Author notes

  1. Anton V. Zabrodin

    Present address: Institute of Chemical Physics, Kosygina Str. 4, SU-117334, Moscow, Russia

Authors and Affiliations

  1. Enrico Fermi Institute, Department of Physics and Mathematical Disciplines Center, University of Chicago, 60637, Chicago, IL

    Peter G. O. Freund & Anton V. Zabrodin

Authors
  1. Peter G. O. Freund
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  2. Anton V. Zabrodin
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Additional information

Communicated by G. Felder

Work supported in part by the NSF: PHY-91-23780

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Freund, P.G.O., Zabrodin, A.V. The spectral problem for theq-Knizhnik-Zamolodchikov equation and continuousq-Jacobi polynomials. Commun.Math. Phys. 173, 17–42 (1995). https://doi.org/10.1007/BF02100180

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  • DOI: https://doi.org/10.1007/BF02100180

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