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

Communications in Mathematical Physics volume 173, pages 17–42 (1995)Cite this article

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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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

Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Asymptotic Behavior
  • Nonlinear Dynamics