Disjoint data inverse problem on manifolds with quantum chaos bounds

We consider the inverse problem to determine a smooth compact Riemannian manifold $(M,g)$ from a restriction of the source-to-solution operator, ${\mathrm{\Lambda }}_{\mathcal{S},\mathcal{R}}$, for the wave equation on the manifold. Here, $\mathcal{S}$ and $\mathcal{R}$ are open sets on $M$, and ${\mathrm{\Lambda }}_{\mathcal{S},\mathcal{R}}$ represents the measurements of waves produced by smooth sources supported on $\mathcal{S}$ and observed on $\mathcal{R}$. We emphasise that $\bar{\mathcal{S}}$ and $\bar{\mathcal{R}}$ could be disjoint. We demonstrate that ${\mathrm{\Lambda }}_{\mathcal{S},\mathcal{R}}$ determines the manifold $(M,g)$ uniquely under the following spectral bound condition for the set $\mathcal{S}$: There exists a constant $C>0$ such that any normalized eigenfunction ${\phi }_k$ of the Laplace-Beltrami operator on $(M,g)$ satisfies \begin{equation*} 1\leq C|φ_k|_{L^2($\mathcal{S}$)}. \end{equation*} We note that, for the Anosov surface, this spectral bound condition is fulfilled for any non-empty open subset $\mathcal{S}$. Our approach is based on the paper [18] and the spectral bound condition above is an analogue of the Hassell-Tao condition there.