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

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