Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms

Abstract

Abstract We consider the inverse problem of determining simultaneously two unknown electric potential coefficients for a system of two general strongly coupled Schrödinger equations, with magnetic potential terms, and with Neumann boundary conditions, from single Dirichlet measurements on a portion Γ1 of the boundary. Under suitable geometrical assumptions on the complementary unobserved portion Γ0 of the boundary, we show that one can uniquely determine the two unknown potential coefficients in one shot, from respective Dirichlet boundary measurements on Γ1 over an arbitrarily short time interval. The proof is based on a recent Carleman estimate in [Lasiecka, Triggiani and Zhang, J. Inv. Ill-Posed Problems 12: 43–123, 2004] for single Schrödinger equations. It also takes advantage of a convenient route “post-Carleman estimates” suggested by [Isakov, Inverse Problems for Partial Differential Equations, Springer, 2006, Theorem 8.2.2, p. 231].