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Adequate observation time on a simple surface

Example

Assume Countable Choice. Let $(M, g)$ be a compact, connected, nonempty simple two-dimensional Riemannian manifold with smooth boundary. Suppose $L_{*} < \infty$ is an upper bound for the lengths of all maximal unit-speed geodesics in $M$ . Let $T > L_{*}$ and let $q_{1}, q_{2} \in C_{c}^{\infty} (M^{\circ}; R)$ be static interior potentials. If their full time-dependent Dirichlet-to-Neumann maps agree for every smooth boundary input compactly supported in $(0, T) \times \partial M$ , then $q_{1} = q_{2}$ throughout $M$ .

This gives a concrete observation-time condition: observing longer than a uniform upper bound on the maximal geodesic lengths suffices for coefficient uniqueness in the simple-surface case.

Example references

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

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 Riemannian surface; a finite uniform upper bound $L_{*}$ on all maximal geodesic lengths; $T > L_{*}$ ; two real smooth interior potentials with equal full time-dependent DN maps.

[A1]

Countable Choice is the axiom base of the finite-time coefficient-uniqueness and two-dimensional X-ray injectivity suppliers. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Under Countable Choice, on a compact connected nonempty simple surface, if $L_{*} < \infty$ bounds all maximal unit-speed geodesic lengths and $T > L_{*}$ , equality of the full forward DN maps on smooth time-compact boundary inputs implies $q_{1} = q_{2}$ for real static $q_{1}, q_{2} \in C_{c}^{\infty} (M^{\circ}; R)$ . The vanishing ray integrals are proved inside this corollary, not required as an additional premise. (Static wave coefficients are determined on a simple surface with adequate observation time)

[F2]

The unweighted scalar geodesic X-ray transform is injective on a compact simple surface. (Unweighted scalar geodesic X-ray injectivity on simple surfaces)

Proof

1.1given

By hypothesis, every maximal unit-speed geodesic has length at most $L_{*} < T$ . Thus all maximal rays relevant to the finite-time coefficient-uniqueness corollary lie within the observation interval; the strict inequality also covers a ray attaining the upper bound.

2.1A1F1F2step 1.1given∎

Equal DN maps and the simple-surface hypotheses satisfy [F1]; since $T > L_{*}$ , the corollary yields $q_{1} - q_{2} = 0$ on $M$ . Its X-ray injectivity step is justified by [F2], so $q_{1} = q_{2}$ as claimed.

Depends on

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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)