Counterexamples to inverse problems for the wave equation

TL;DR

This paper constructs counterexamples demonstrating that inverse problems for the wave equation can have non-unique solutions, especially in Lorentzian manifolds, challenging the assumptions of uniqueness in inverse problems.

Contribution

It provides explicit counterexamples of non-uniqueness in inverse wave problems on Lorentzian manifolds and Euclidean domains, highlighting limitations of current inverse problem techniques.

Findings

Non-isometric Lorentzian metrics can produce identical partial data measurements.

Counterexamples involve smooth, non-degenerate, time-dependent metrics.

Metrics conformal to Minkowski metric on Euclidean domains.

Abstract

We construct counterexamples to inverse problems for the wave operator on domains in $\mathbb{R}^{n+1}$, $n \ge 2$, 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 $\mathbb{R}^{n+1}$ the metrics are conformal to the Minkowski metric.