Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights

Alberto Mercado (1) , Axel Osses (2) , Lionel Rosier (3)

1 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]
2 DIMEC - Departamento de Ingeniería Mecánica
3 IECN - Institut Élie Cartan de Nancy

Baudouin and Puel (2002 Inverse Problems 18 1537-54), investigated some inverse problems for the evolution Schr¨odinger equation bymeans of Carleman inequalities proved under a strict pseudoconvexity condition. We showhere that new Carleman inequalities for the Schr¨odinger equationmay 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 RN. 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 Schr¨odinger equation from a single boundary or internal measurement.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Alberto Mercado, Axel Osses, Lionel Rosier. Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights. Inverse Problems, 2008, 24 (1), pp.1-18. ⟨10.1088/0266-5611/24/1/015017⟩. ⟨hal-00600940⟩