Stability estimate for an inverse wave equation and a multidimensional Borg–Levinson theorem

We consider the stability in an inverse problem of determining the potential q entering the wave equation in a bounded smooth domain of Rd from boundary observations. The observation is given by a hyperbolic (dynamic) Dirichlet to Neumann map associated to a wave equation. We prove a log-type stability estimate in determining q from a partial Dirichlet to Neumann map provided that q is a priori known in a neighbourhood of the boundary of the spatial domain and satisfies an additional condition. Next, we use this result to establish a stability estimate related to the multidimensional Borg–Levinson theorem.