A Generalized Hypergeometric Function III. Associated Hilbert Space Transform

Abstract

For generic parameters (a ₊ ,a ₋ ,c)∈(0,∞)²×ℝ⁴, we associate a Hilbert space transform to the ‘‘relativistic’’ hypergeometric function $R({a_{+},a_{-}},{\bf c};v,\hat{v})$ studied in previous papers. Restricting the couplings c to a certain polytope, we show that the (renormalized) R-function kernel gives rise to an isometry from the even subspace of $L^2({{\mathbb R}},\hat{w}(\hat{v})d\hat{v})$ to the even subspace of L ²(ℝ,w(v)dv), where $\hat{w}(\hat{v})$ and w(v) are positive and even weight functions. We prove that the orthogonal complement of the range of this isometry is spanned by N∈ℕ pairwise orthogonal functions. The latter are in essence Askey-Wilson polynomials, arising from the R-function by choosing $\hat{v}=i\kappa_n$, with ${{\kappa_0,\ldots,\kappa_{{N-1}}}}$ distinct negative numbers. The two commuting analytic difference operators acting on the variable v for which R is a joint eigenfunction, give rise to two commuting self-adjoint Hamiltonians on the even subspace of L ²(ℝ,w(v)dv). We explicitly determine the relation of the time-dependent scattering theory for these dynamics to their joint spectral transform.