The Spectral Problem for the q-Knizhnik-Zamolodchikov Equation and Continuous q-Jacobi Polynomials

The spectral problem for the q-Knizhnik-Zamolodchikov equations for $U_{q}(\widehat{sl_2}) (0<q<1)$ at arbitrary 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 shown to coincide, up to a trivial factor, with the kink-antikink $S$-matrix for the spin-$\frac{1}{2}$ XXZ antiferromagnet. In the limit of infinite level we observe connections with harmonic analysis on $p$-adic groups with the prime $p$ given by $p=q^{-2}$.