Carleman inequalities and inverse problems for the Schrödinger equation (original) (raw)
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, investigated some inverse problems for the evolution Schrödinger equation by means of Carleman inequalities proved under a strict pseudoconvexity condition. We show here that new Carleman inequalities for the Schrödinger equation may be derived under a relaxed pseudoconvexity condition, which allows us to use degenerate weights with a spatial dependence of the type ψ(x) = x · e, where e is some fixed direction in R N . As a result, less restrictive boundary or internal observations are allowed to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from a single boundary or internal measurement.
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Journal de Mathématiques Pures et Appliquées, 2009
In this paper, we investigate the inverse problem of determining the potential of the dynamical Schrödinger equation in a bounded domain from the data of the solution in a subboundary over a time interval. Assuming that in a neighborhood of the boundary of the spatial domain, the potential is known and without any assumption on the dynamics (i.e. without the geometric optics condition for the observability), we prove a logarithmic stability estimate for the inverse problem with a single measurement on an arbitrarily given subboundary.
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Comptes Rendus Mathematique
We establish geometrical conditions for the inverse problem of determining a stationary potential in the wave equation with Dirichlet data from a Neumann measurement on a suitable part of the boundary. We present the stability results when we measure on a part of the boundary satisfying a rotated exit condition. The proofs rely on global Carleman estimates with angle type dependence in the weight functions.