The scattering matrix for the p-form Laplacian on asymptotically conic manifolds
Abstract
The Hodge decomposition at infinity gives rise to distinct tangential scattering channels for differential forms on asymptotically conic manifolds. We construct co-closed generalized eigenforms for the Hodge Laplacian on p-forms in dimension n≥ 3 and analyze the resulting Bessel and Hankel systems. From these we determine the scattering matrix modulo smoothing operators, identifying it as a Fourier integral operator associated to the time π geodesic flow and whose principal symbol encodes fibre-wise parallel transport with respect to the Levi-Civita connection on differential forms. In this setting this lifts the main result of Melrose and Zworski in [24] to differential forms and identifies additional scattering phase. The construction also yields a spectral representation in the co-closed sector, including a Stone-type formula and a functional calculus. In dimension $3$ and degree p=1, this framework describes divergence-free fields and thus the electric-field component of Maxwell's equations.
Related
- Strichartz and Spectral Projection Estimates on Asymptotically Conic Manifolds
- Carleman Estimates for Critical Electromagnetic Schrödinger Operators on Conic Manifolds
- Positive mass theorems for singular asymptotically hyperbolic manifolds
- A new boundary mass for asymptotically flat half-manifolds
- Convex hulls and area minimizing currents in asymptotically locally hyperbolic manifolds
BibTeX
@misc{charalambous2026scattering,
title = {{The scattering matrix for the p-form Laplacian on asymptotically conic manifolds}},
author = {Nelia Charalambous and Alden Waters},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22984},
eprint = {2608.22984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22984},
}
RIS
TY - UNPB
TI - The scattering matrix for the p-form Laplacian on asymptotically conic manifolds
AU - Nelia Charalambous
AU - Alden Waters
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2608.22984
UR - https://arxiv.org/abs/2608.22984
AB - The Hodge decomposition at infinity gives rise to distinct tangential scattering channels for differential forms on asymptotically conic manifolds. We construct co-closed generalized eigenforms for the Hodge Laplacian on p-forms in dimension n≥ 3 and analyze the resulting Bessel and Hankel systems. From these we determine the scattering matrix modulo smoothing operators, identifying it as a Fourier integral operator associated to the time π geodesic flow and whose principal symbol encodes fibre-wise parallel transport with respect to the Levi-Civita connection on differential forms. In this setting this lifts the main result of Melrose and Zworski in [24] to differential forms and identifies additional scattering phase. The construction also yields a spectral representation in the co-closed sector, including a Stone-type formula and a functional calculus. In dimension $3$ and degree p=1, this framework describes divergence-free fields and thus the electric-field component of Maxwell's equations.
ER -