The purpose of this article is to define the radiation fields on asymptotically hyperbolic manifolds and to use them to study scattering theory. The radiation fields on R n and on asymptotically Euclidean manifolds were introduced by F.G. Friedlander in a series of papers starting in the early 1960’s [10, 11, 12, 13, 14]. His program of using the radiation fields to obtain the scattering matrix in that general setting was

Introduction The theory of Hilbert spaces of entire functions [1] was developed by Louis de Branges in the late 1950s and early 1960s with the help of his students including James Rovnyak and David Trutt. It is a generalization of the part of Fourier analysis involving Fourier transform and Plancherel formula. The de Branges-Rovnyak theory of square summable power series, whichplayed an important role in leading to de Branges' discovery of a proof of the Bieberbach conjecture, originated from the theory of Hilbert spaces of entire functions. In [2] de Branges proposed an approach to the generalized Riemann hypothesis, that is, the hypothesis that not only the Riemann zeta function i(s) but also all the Dirichlet L-functions L(s# ) with primitive have their nontrivial zeros lying on the critical line !s = 1=2 (See Davenport [5]). In [2] de Br