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, Λ _S,R for the wave equation on the manifold. Here, S and R are open sets on M, and Λ _S,R represents the measurements of waves produced by smooth sources supported on S and observed on R . We emphasize that S and R could be disjoint. We demonstrate that Λ _S,R determines the manifold (M, g) uniquely under the following spectral bound condition for the set S : There exists a constant C > 0 such that any normalized eigenfunction φ of the Laplace-Beltrami operator on (M, g) satisfies 1 ≤ C|| φ | S | L2(S). We note that, for the Anosov surface, this spectral bound condition is fulfilled for any nonempty open subset S. Moreover, we solve the analogue of this problem for the heat equation by showing that the source-to-solution maps for the heat and wave equations determine each other.