Stable determination of coefficients in the dynamical anisotropic Schr{ö}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 Schrödinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension n bigger than 2 that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the electric potential and we establish Hölder-type stability estimates in determining the potential. We prove similar results for the determination of a conformal factor close to 1.

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