Formulas for \(q\)-spherical functions using inverse scattering theory of reflectionless Jacobi operators (Q5938840)

The authors study the spectral problem for a Ruijsenaars-type difference (\(q\)-) version of the one-dimensional Schrödinger operator with Pöschl-Teller potential. In the classical case, the eigenfunctions occur as zonal spherical functions on hyperboloids, and it is argued that in the \(q\)-case there should be an analogous interpretation on quantum hyperboloids. Using inverse scattering theory for Jacobi operators, the authors derive a Sato-type combinatorial formula for the eigenvectors. It can be written in terms of Schur functions, which yields a connection with characters of \(\text{SL}(N,\mathbb{C})\). The authors also give a more compact expression as a terminating \({}_2\varphi_1\) basic hypergeometric sum.