Uniqueness in an inverse problem for a system of coupled Schrödinger equations with Dirichlet-Neumann boundary Conditions

• Published in: STUDIES IN ENGINEERING AND EXACT SCIENCES Vol. 5; no. 2; p. e11620
• Main Author: Atef, Saci
• Format: Journal Article
• Language: English
• Published: 05.12.2024
• ISSN: 2764-0981; 2764-0981
• DOI: 10.54021/seesv5n2-639

When modeling a phenomenon using partial differential equations, the physical/ mechanical/ biological parameters involved are not necessarily well known. However, the resolution of these equations, which is the subject of the direct problem, can only be done if all the data of the system are identified (initial and boundary conditions, coefficients involved in the equations, spatial domain, etc.). If this is not the case, additional information, via experimental measurements for example, is then necessary to determine them. The mathematical notion of an inverse problem consists of the possibility of finding the value of a parameter from partial measurements (localized, for a given time, possibly repeated) on the solution of the system considered. This article concerns the inverse problem of the recovery of two unknown potential coeffcients for a coupled system of two Schrödinger equations, in a bounded domain of  with Dirichlet-Neumann boundary conditions from a Neumann-Dirichlet boundary measurement. We prove uniqueness for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and under weak regularity requirements on the data. Our proof relies on sharp Carleman estimates in (LASIECKA et al., 2004) for Schrödinger equations.