CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Static wave coefficients are determined on a simple surface with adequate observation time

Statement

Assume Countable Choice. Let $(M, g)$ be a compact, connected, nonempty simple two-dimensional Riemannian background with smooth boundary. Suppose $L_{*} < \infty$ bounds the lengths of all maximal unit-speed geodesics in $M$ , and let $T > L_{*}$ . For real static potentials $q_{1}, q_{2} \in C_{c}^{\infty} (M^{\circ}; R)$ , if the full time-dependent Dirichlet-to-Neumann maps for $\partial_{t}^{2} - Δ_{g} + q_{1}$ and $\partial_{t}^{2} - Δ_{g} + q_{2}$ agree for every smooth boundary input compactly supported in $(0, T) \times \partial M$ , then $q_{1} = q_{2}$ on $M$ . This equality statement asserts no stability modulus or recovery in the presence of noisy data.

Statement references

Wave boundary data determine integrals along maximal geodesics shorter than T

Simple Riemannian background

Simple surface and geodesic X-ray transform

Unweighted scalar geodesic X-ray injectivity on simple surfaces

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: Countable Choice; a simple compact connected nonempty Riemannian surface; a finite upper bound $L_{*}$ on all maximal unit-speed geodesic lengths; $T > L_{*}$ ; real smooth interior potentials with equal full time-dependent DN maps.

[A1]

Countable Choice is the axiom base of the wave-data and two-dimensional X-ray suppliers. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Equal wave DN maps on a simple background imply that the integral of $q_{1} - q_{2}$ over every maximal unit-speed geodesic of length less than $T$ is zero. (Wave boundary data determine integrals along maximal geodesics shorter than T)

[F2]

The scalar geodesic X-ray transform integrates a real interior function over each incoming maximal unit-speed boundary geodesic. (Simple surface and geodesic X-ray transform)

[F3]

On a compact simple surface, zero unweighted scalar geodesic X-ray data for a real smooth interior function imply that the function vanishes. (Unweighted scalar geodesic X-ray injectivity on simple surfaces)

Proof

1.1A1F1F2given

Every maximal unit-speed geodesic has length at most $L_{*} < T$ . Therefore [F1] applies to each such ray and gives $I_{0} (q_{1} - q_{2}) = 0$ on all incoming boundary unit vectors, using the transform convention in [F2].

2.1A1F3step 1.1given∎

The real function $q_{1} - q_{2}$ lies in the class covered by [F3]. Its transform vanishes by step 1.1, so [F3] gives $q_{1} - q_{2} = 0$ throughout $M$ . Hence $q_{1} = q_{2}$ .

Depends on

Used by

Adequate observation time on a simple surface Example

Dependency tree · two levels

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Sources

Mikko Salo, On geometric inverse problems and microlocal analysis (full author lecture notes, arXiv:2312.08123v1) (standard reference, not scraped)
Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann, Geometric inverse problems, with emphasis on two dimensions (2022 monograph) (standard reference, not scraped)