An inverse problem for Schrodinger equations with discontinuous main coefficient

This article concerns the inverse problem of retrieving a stationary potential for the Schrodinger evolution equation in a bounded domain of N with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and Lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and on the sign of the jump of a at the interface. The proof is based on a global Carleman inequality for the Schrodinger equation with discontinuous coefficients, result also interesting by itself.