From Penrose to Melrose: Computing Scattering Amplitudes at Infinity for Unbounded Media

We develop a method to compute scattering amplitudes for the Helmholtz equation in variable, unbounded media with possibly long-range asymptotics. Combining Penrose's conformal compactification and Melrose's geometric scattering theory, we formulate the time-harmonic scattering problem on a compactified manifold with boundary and construct a two-step solver for scattering amplitudes at infinity. The construction is asymptotic: it treats a neighborhood of infinity, and is meant to couple to interior solvers via domain decomposition. The method provides far-field data without relying on explicit solutions or Green's function representation. Scattering in variable media is treated in a unified framework where both the incident and scattered fields solve the same background Helmholtz operator. Numerical experiments for constant, short-range, and long-range media with single-mode and Gaussian beam incidence demonstrate spectral convergence of the computed scattering amplitudes in all cases.