Recovery of a matrix valued potential for the wave equation on stationary spacetimes

2025/10/15 by Spyridon Filippas, Lauri Oksanen, Filippas, Spyridon +3 · 1 citation
Mathematics · Physics and Astronomy · #35R30 #53C50 #53C65 #Analysis of PDEs (math.AP) #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #Relativity and Gravitational Theory #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2510.13410

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

We study the problem of recovering a time dependent matrix valued potential on a globally hyperbolic manifold from the knowledge of the source to solution map of a wave equation including a connection 1-form term. We exhibit sufficient conditions for solving this inverse problem under the assumption that the the manifold is stationary and that the connection term is time independent. The proof is based on two ingredients. The first is reduction of the problem to the study of a non-Abelian light ray transform and holds assuming global hyperbolicity only. The second is the study of this transform and establishing a link with a Riemannian analogue.

Citations

• On the interplay between the light ray and the magnetic X-ray transforms
• Rigidity in the Lorentzian Calderón problem with formally determined data
• Stable determination of a time-dependent matrix potential for a wave equation in an infinite waveguide
• Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients
• Lorentzian Calderón problem under curvature bounds
• Recovery of a time-dependent Hermitian connection and potential\n appearing in the dynamic Schr "odinger equation
• Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations
• The geodesic X-ray transform with matrix weights
• Inverse problems for the connection Laplacian
• The Attenuated Magnetic Ray Transform on Surfaces
• The boundary rigidity problem in the presence of a magnetic field
• Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials

Cited by

• The Lorentzian Calderón problem on vector bundles

Related

• Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations
• Inverse problem for connections in semi-linear wave equations on Lorentzian manifolds
• Recovery of a time-dependent potential in hyperbolic equations on conformally transversally anisotropic manifolds
• Stable determination of a time-dependent matrix potential for a wave equation in an infinite waveguide
• On Solutions to the Wave Equation on Non-globally Hyperbolic Manifold