Stability estimate for an inverse problem for the electro-magnetic wave equation and spectral boundary value problem

In this paper, we prove the stability estimate of the inverse problem for the determination of the electric potential using the Neumann spectral data {lambda(k), partial derivative(nu)phi(k), k >= 1}. The uniqueness result is given in Katchalov and Kurylev (1998 Commun. Partial Differ. Equ. 23 55-95), where the authors show that the canonical Schrodinger operator is uniquely determined via its incomplete boundary spectral data. To obtain this result, we establish the stability estimate of the inverse problem of determining the electric potential entering the electro-magnetic wave equation in a bounded smooth domain in R-d from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated with the solutions to the electro-magnetic wave equation. We prove in dimension d >= 2 that the knowledge of the Dirichlet-to-Neumann map for the electro-magnetic wave equation measured on the boundary uniquely determines the electric potential.