Hyperbolic inverse problems and finite-time visibility — Examples and Counterexamples

Let $(M,g)$ be a compact Riemannian manifold, let $T>0$, and let $Q=(0,T)\times M$ carry product measure $d{V}_{g}dt$. Fix a unit-speed geodesic segment $\gamma :[0,L]\to M$ and $t_0$ with $0<t_0<T-L$, and set $\mathrm{\Gamma }(s)=(t_0+s,\gamma (s))$, $0\le s\le L$; $ds$ denotes its arclength parameter. Let $a\in C^{\mathrm{\infty }}([0,T]\times M)$, and for each $\lambda \ge 1$ let $u_{j,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}\in L^2(Q)$ and $R_{j,\lambda }\in L^2(Q)$ for $j=1,2$. Suppose $\mathrm{\parallel }{u}_{j,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}{\mathrm{\parallel }}_{L^2(Q)}\le C_0$, $\mathrm{\parallel }{R}_{j,\lambda }{\mathrm{\parallel }}_{L^2(Q)}\le C_1{\lambda }^{-1\mathrm{/}2}$, and ${\int }_{Q}a{u}_{1,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}{u}_{2,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}d{V}_{g}dt⟶{\int }_0^{L}a(t_0+s,\gamma (s))ds\mathrm{.}$ Set $u_{j,\lambda }=u_{j,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}+R_{j,\lambda }$. Then $\mid {\int }_{Q}a{u}_{1,\lambda }{u}_{2,\lambda }d{V}_{g}dt-{\int }_{Q}a{u}_{1,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}{u}_{2,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}d{V}_{g}dt\mid =O({\lambda }^{-1\mathrm{/}2}),$ and hence the exact-solution product has the same ray-integral limit. For the real self-adjoint wave application, assume additionally Countable Choice, that $M$ is connected and nonempty with nonempty smooth boundary, and that the two operators are $P_j={\mathrm{\partial }}_t^2-{\mathrm{\Delta }}_g+q_j(x)$ with $q_j\in C^{\mathrm{\infty }}(M;\mathbb{R})$ independent of time. The beams must satisfy the following additional supplier conditions: the observed ray has a compact embedded phase-space extension with immersed base projection; its two endpoint cutoff neighborhoods lie outside the measurement cylinder; and the approximate forward beam vanishes near $t=0$, while the approximate terminal beam vanishes near $t=T$. These temporal vanishing conditions are separate from the location of the spatial endpoint residuals. Use physical time along the unit-speed spatial ray and the normalization $\mathrm{\mid }{\mathrm{\partial }}_t\varphi \mathrm{\mid }=1$, with initial transported mass one. Then the Gaussian-beam lemma gives the unweighted ray measure and residual $O({\lambda }^{-N})$ throughout $Q$. Apply the wave energy wellposedness theorem with zero boundary data and zero initial data for the forward correction, or zero terminal data for the terminal correction, to the negative residual. Since $\mathrm{\parallel }{P}_j{u}_{j,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}{\mathrm{\parallel }}_{L_t^1{L}_x^2}\le \sqrt{T}\mathrm{\parallel }{P}_j{u}_{j,\lambda }^{\mathrm{a}\mathrm{p}\mathrm{p}}{\mathrm{\parallel }}_{L^2(Q)}$, its energy estimate gives an $L^2(Q)$ correction $O({\lambda }^{-N})$, hence the assumed $O({\lambda }^{-1\mathrm{/}2})$ order after choosing $N\ge 1$. The exact waves retain the actual approximate boundary traces. This application asserts no temporal vanishing merely from an extension outside the measurement domain; the generic error estimate above only needs its explicitly stated norm and product-limit hypotheses.

Gaussian beams concentrated on an embedded null bicharacteristic

Energy well-posedness for the variable-metric wave equation

Holder's inequality for integrals, including the endpoint cases

Riemannian metric and riemannian manifold

Riemannian volume density

Riemannian speed and length

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Assume Countable Choice. Let $(M,g)$ be a compact, connected, nonempty smooth Riemannian manifold with nonempty smooth boundary, let $T>0$, and let $q_1,q_2\in C^{\mathrm{\infty }}(M;\mathbb{R})$ be time independent. Put $Q_0=\mathrm{supp}(q_1-q_2)$, the closure of the set on which $q_1-q_2$ is nonzero, and suppose $Q_0\mathrm{\ne }\mathrm{\varnothing }$ and $d_g(Q_0,\mathrm{\partial }M)>T\mathrm{/}2$, where $d_g$ is the intrinsic Riemannian distance. For every smooth Dirichlet input $h\in C_c^{\mathrm{\infty }}((0,T)\times \mathrm{\partial }M)$, the full time-dependent forward Dirichlet-to-Neumann maps satisfy ${\mathrm{\Lambda }}_{q_1}^{+}h={\mathrm{\Lambda }}_{q_2}^{+}h\text{in }{L}^2((0,T)\times \mathrm{\partial }M,d{S}_{g}dt)\mathrm{.}$ Thus a perturbation supported farther than half the observation time from the boundary cannot be detected by boundary measurements made before time $T$.

Finite propagation for the wave equation on a Riemannian manifold

Smooth boundary tests separate locally integrable traces

Energy well-posedness for the variable-metric wave equation

Time-dependent Dirichlet-to-Neumann map for the wave equation

Alessandrini identity for the time-dependent wave Dirichlet-to-Neumann maps

Riemannian distance on a connected manifold

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Assume Countable Choice. Let $(M,g)$ be a compact, connected, nonempty simple two-dimensional Riemannian manifold with smooth boundary. Suppose $L_{\ast }<\mathrm{\infty }$ is an upper bound for the lengths of all maximal unit-speed geodesics in $M$. Let $T>L_{\ast }$ and let $q_1,q_2\in C_c^{\mathrm{\infty }}(M^{\circ };\mathbb{R})$ be static interior potentials. If their full time-dependent Dirichlet-to-Neumann maps agree for every smooth boundary input compactly supported in $(0,T)\times \mathrm{\partial }M$, 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.

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 (${\mathrm{A}\mathrm{C}}_{\omega }$)