An inverse problem for Schrödinger equations with discontinuous main coefficient

Abstract

This article concerns the inverse problem of retrieving a stationary potential for the Schrödinger evolution equation in a bounded domain of ℝ ^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 Schrödinger equation with discontinuous coefficients, result also interesting by itself.

Keywords:

• inverse problem
• Carleman inequality
• degenerate weight
• pseudoconvexity
• Schrödinger equation

AMS Subject Classifications: :

• 35R30
• 35Q40

Acknowledgements

A. Mercado would like to acknowledge the partial support by ECOS grant C04E08 and NSF grant DMS 0554571.