An inverse problem for Schr\"odinger equations with discontinuous main coefficient

TL;DR

This paper addresses the inverse problem of determining a stationary potential in a Schrödinger equation with a discontinuous coefficient, establishing uniqueness and stability results using Carleman inequalities under geometric conditions.

Contribution

It introduces a novel approach to inverse Schrödinger problems with discontinuous coefficients, proving Lipschitz stability and uniqueness with new Carleman estimates.

Findings

Proved uniqueness of the potential reconstruction.

Established Lipschitz stability under geometric assumptions.

Developed a global Carleman inequality for discontinuous coefficients.

Abstract

This paper concerns the inverse problem of retrieving a stationary potential for the Schr\"odinger evolution equation in a bounded domain of RN 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 Schr\"odinger equation with discontinuous coefficients, result also interesting by itself.