Two theorems show that a potential on a closed Riemannian manifold is uniquely recovered from sparse eigenvalue data and eigenfunction restrictions to an open set, with applications to single-measurement passive imaging.
Semiclassical measures for complex hyperbolic quotients
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We study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold. The proof uses the one-dimensional fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] along the fast expanding/contracting directions, in a way similar to the work of Dyatlov-J\'ez\'equel [arXiv:2108.10463] in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.
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Inverse spectral problems with sparse data and applications to passive imaging on manifolds
Two theorems show that a potential on a closed Riemannian manifold is uniquely recovered from sparse eigenvalue data and eigenfunction restrictions to an open set, with applications to single-measurement passive imaging.