2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68897We exhibit the form of the ``radiation field,'' describing the large-scale, long-time behavior of solutions to the wave equation on a manifold with no trapped rays, as a Fourier integral operator. We work in two different geometric settings: scattering manifolds (a class which includes asymptotically Euclidean spaces) and asymptotically hyperbolic manifolds. The canonical relation of the radiation field operator is a map from the cotangent bundle of the manifold to a cotangent bundle over the boundary at infinity; it is associated to a sojourn time, or Busemann function, for geodesic rays. In non-degenerate cases, the symbol of the operator can be described explicitly in terms of the geometry of long-time geodesic flow. As a consequence of the above result, we obtain a description of the (distributional) high-frequency asymptotics of the scattering-theoretic Poisson operator, better known as the Eisenstein function in the asymptotically hyperbolic case.Analysis of PDEs35L05; 58J45; 58J40; 58J50text