From Penrose to Melrose: Computing Scattering Amplitudes at Infinity for Unbounded Media
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917188561010688 |
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| author | Zenginoğlu, Anıl |
| author_facet | Zenginoğlu, Anıl |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Penrose to Melrose: Computing Scattering Amplitudes at Infinity for Unbounded Media Zenginoğlu, Anıl Numerical Analysis General Relativity and Quantum Cosmology 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. |
| title | From Penrose to Melrose: Computing Scattering Amplitudes at Infinity for Unbounded Media |
| topic | Numerical Analysis General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2601.04167 |