Unique continuation for general non-trapping obstacles and trajectories

PDF p. 21: §5.2(A)

Assume the Axiom of Choice: every family of nonempty sets admits a choice function. Let $n\ge 2$, $T>0$, and $\lambda \in \mathbb{R}\setminus \{0\}$. Let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty compact obstacle with smooth boundary, connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$, and the non-trapping property of Non-trapping obstacle. For any $R>0$ with $\mathrm{\Theta }\subset B_R(0)$, put $\omega =\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }>R\}$.

For every reference trajectory $w$ satisfying (W) as defined in Reference trajectory hypothesis W, does (UCPw), as defined in Conditional energy-level unique continuation hypothesis, hold for this $T$ and $\omega$? Explicitly, does every $v\in C([0,T];H_0^1(\mathrm{\Omega }))$ satisfying $i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+2\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}v-\lambda w^2\bar{v}=0$ in distributions on $(0,T)\times \mathrm{\Omega }$ and vanishing almost everywhere on $(0,T)\times \omega$ vanish identically as an $H_0^1(\mathrm{\Omega })$-valued path?

The frozen article, §5.2(A), reports this question open for general trajectories and non-trapping obstacles. It notes that the coefficients can depend on time and need not be analytic, and proposes Carleman estimates as a possible approach while noting that known approaches commonly impose additional geometry. This is attribution of the authors' status and suggested approach, not a proof that the question remains open worldwide.

A related result of Filippas, Laurent and Léautaud treats Schrödinger operators with lower-order coefficients Gevrey 2 in time; their Theorem 1.7 concerns a scalar equation on the unit disk under additional hypotheses. These conditions and that scalar setting do not establish the question above for the conjugate-coupled equation and arbitrary (W) trajectories. A limited literature search on 2026-11-13 did not identify a solution covering the complete stated formulation; it is not an exhaustive current-status review.