Counterexamples to inverse problems for the wave equation

Abstract

We construct counterexamples to inverse problems for the wave operator on domains in \begin{document}$ \mathbb{R}^{n+1} $\end{document}, \begin{document}$ n \ge 2 $\end{document}, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On \begin{document}$ \mathbb{R}^{n+1} $\end{document} the metrics are conformal to the Minkowski metric.