Stable determination of coefficients in the dynamical anisotropic Schr{\'o}dinger equation from the Dirichlet-to-Neumann map

In this paper we are interested in establishing stability estimates in the inverse problem of determining on a compact Riemannian manifold the electric potential or the conformal factor in a Schr\"odinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the Schr\"odinger equation. We prove in dimension n bigger than 2 that the knowledge of the Dirichlet-to-Neumann map for the Schr\"odinger equation uniquely determines the electric potential and we establish H\"older-type stability estimates in determining the potential. We prove similar results for the determination of a conformal factor close to 1.