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

TL;DR

This work introduces null infinity compactification (NIC) to compute Helmholtz scattering amplitudes in unbounded, variable media by merging Penrose conformal ideas with Melrose scattering calculus. A two-step solver prescribes incoming data at past null infinity $\mathscr I^-$, computes the corresponding incident field, then solves for the scattered field to read off the outgoing data at future null infinity $\mathscr I^+$, with a WKB-type rescaling and radial compactification that regularize infinity. The method yields spectrally accurate far-field amplitudes for constant, short-range, and Coulomb-type long-range media, including non-radial anisotropic cases on hyperbolic geometries, and demonstrates stable coupling to interior solvers via domain decomposition. By embedding infinity into the computational domain, NIC provides a geometric, operator-theoretic foundation for scalable wave scattering computations in unbounded domains and offers a path toward three-dimensional and Maxwell-system extensions.

Abstract

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.

Figures (11)

• Figure 1: We solve the time-harmonic scattering problem \ref{['eq:helmholtz']}, where incoming radiation from infinity, $u^-_\infty(\omega)$ is scattered by an obstacle $\mathcal{D}$ and is measured at infinity as the far field $u^+_\infty(\omega)$. The computation consists of two steps based on the separation of the total field $U$ into incident $(-)$ and scattered $(+)$ fields: $U=U^-+U^+$. In the first step, we compute the incident field $U^-$ by prescribing incoming boundary conditions at past null infinity, $\mathscr{I}^{-}$, and extract the data on the obstacle boundary $\Gamma=\partial\mathcal{D}$. In the second step, we use the obstacle boundary data to solve for the scattered field $U^+$ and read off the far field at future null infinity, $\mathscr{I}^{+}$. The singularity of the map between $U^\pm$ and $u^\pm_\infty$ is absorbed by a suitable weight function \ref{['eq:scale_out']} that corresponds to regular compactification at null infinity $\mathscr I^\pm$ described in Sec. \ref{['sec:penrose']}.
• Figure 2: Real parts on a shared axis for the two types of incoming data, $\Re u_\infty^-(\theta)$, prescribed at past null infinity $\mathscr I^-$: single mode \ref{['eq:single-mode']} with $m=8$ and a circular Gaussian beam using a von Mises distribution \ref{['eq:gaussian']} with $\beta=8, \ \theta_0=\pi$.
• Figure 3: Relative errors in the discrete $\ell^2$ norm of incident (solid) and scattered (dashed) fields vs. wavenumber $k$ for four azimuthal indices $m\in\{0,4,8,16\}$ in two resolutions $N=\{16,64\}$ using a 1D Chebyshev spectral method. The vertical dashed lines mark the centrifugal-barrier. The scattered field is systematically more accurate; incident errors are high below the barrier but drop rapidly beyond.
• Figure 4: The rescaled incident and scattered fields, $u^\pm$, for single-mode incoming radiation with $m=40$ and $k=120$. The inner circle is the obstacle surface; the outer circle is null infinity (compare Fig. \ref{['fig:incident-scattered']}). The solution is dominated by the angular oscillation with relatively straight rays connecting the obstacle with infinity. The scattering map is unitary (pure phase).
• Figure 5: Radial spectral convergence of the modal scattering map at null infinity. We plot on a semilog scale the absolute error $|S_m(k)-S_m^{\mathrm{num}}(k)|$ against the radial polynomial degree $N_\rho$ for two wavenumbers (left: $k=120$; right: $k=12$) using the same set of ratios $m/k$, and fixed angular resolution $N_\theta=2m+5$. For all cases the error decays exponentially with $N_\rho$ until it reaches saturation. The saturation level and the total error increase with $m/k$ and $k$.

Theorems & Definitions (2)

• Theorem
• Example : Standard hyperboloids