The spectral problem for the \(q\)-Knizhnik-Zamolodchikov equation and continuous \(q\)-Jacobi polynomials. (English) Zbl 0832.35106

Freund, Peter G. O.; Zabrodin, Anton V.

Commun. Math. Phys. 173, No. 1, 17-42 (1995).

Summary: The spectral problem for the \(q\)-Knizhnik-Zamolodchikov equations for \(U_q(\widehat{sl_2})\) \((0< q< 1)\) at arbitrary non-negative level \(k\) 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 continuous \(q\)-Jacobi polynomials, and the \(S\)- matrix is derived from their asymptotic behavior. The level zero \(S\)- matrix is closely connected with the kink-antikink \(S\)-matrix for the spin-\({1\over 2}\) 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 on \(p\)-adic groups with the prime \(p\) given by \(p= q^{- 2}\).

MSC:

35P25	Scattering theory for PDEs
82B23	Exactly solvable models; Bethe ansatz
35Q58	Other completely integrable PDE (MSC2000)
81U20	\(S\)-matrix theory, etc. in quantum theory
81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
33D45	Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

Keywords:

\(q\)-Knizhnik-Zamolodchikov equations; scattering states; \(q\)-Jacobi polynomials; \(S\)-matrix; kink-antikink \(S\)-matrix; harmonic analysis on \(p\)-adic groups

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