Inverse problems for semilinear Schr\'odinger equations at large frequency via polynomial resolvent estimates on manifolds

We study inverse boundary problems for semilinear Schr\"odinger equations on smooth compact Riemannian manifolds of dimensions $\ge 2$ with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities are determined uniquely from the knowledge of the nonlinear Dirichlet--to--Neumann map at a large fixed frequency on quite general Riemannian manifolds. In particular, in contrast to the previous results available, here the manifolds need not satisfy any product structure, may have trapped geodesics, and the geodesic ray transform need not be injective. Only a mild assumption about the geometry of intersecting geodesics is required. We also establish a polynomial resolvent estimate for the Laplacian on an arbitrary smooth compact Riemannian manifold without boundary, valid for most frequencies. This estimate, along with the invariant construction of Gaussian beam quasimodes with uniform bounds for underlying constants and a stationary phase lemma with explicit control over all involved constants, constitutes the key elements in proving the uniqueness results for the considered inverse problems.