Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold

We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set $S$, and the resulting waves are observed on an open receiver set $R$, which may be at a positive distance from $S$. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: $\mathrm{\parallel }\phi {\mathrm{\mid }}_R{\mathrm{\parallel }}_{L^2(R)}\ge c{e}^{-C{\lambda }^{\theta }}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^2(M)}$ for every eigenfunction $\phi$, where $\lambda$ is the corresponding eigenvalue, $\theta \in [0,1\mathrm{/}2)$, and the constants $c,C>0$ do not depend on $\phi$. If a bound of this type holds also on $S$, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with $\theta =1\mathrm{/}2$. Uniqueness nevertheless fails at this endpoint: in every dimension $n\ge 2$, we construct a pair of nonisometric connected, closed Riemannian $n$-manifolds that produce identical wave observations.