||The Schroedinger operator with -h(h+1)/cosh^2 x (h>0) potential is exactly solvable. That is, all the discrete eigenvalues, eigenfunctions and the scattering amplitudes
are exactly calculable. For the latter, the connection formula of the Gaussian hypergeometric function is used. For positive integer h, the potential is ``reflectionless."
The reflectionless potentials of Schroedinger equations are the profiles of KdV solitons. My talk is the difference equation version of these well-known results. The corresponding eigenfunctions consists of the q-ultraspherical polynomials with |q|=1. In order to obtain the scattering amplitudes, the q-ultraspherical polynomials are analytically continued to Heine's q-hypergeometric functions. But their connection formula for |q|=1 is Not known. Based on the conjectured connection formula, the scattering amplitudes