Global recovery of a time-dependent coefficient for the wave equation from a single measurement

We consider the formally determined inverse problem of recovering an unknown time-dependent potential function from the knowledge of the restriction of the solution of the wave equation to a small subset, subject to a single external source. We show that one can determine the potential function, up to the natural obstruction for the problem, by using a single source placed in the exterior of the spacetime domain and subsequently measuring the solution in a small neighborhood outside of the spacetime domain. The approach is based on considering a dense collection of light rays and constructing a source function that combines a countable collection of sources that each generates a wave packet near a light ray in the collection. We show that measuring the solution corresponding to that single source simultaneously determines the light ray transform along all the light rays in the collection. The result then follows from injectivity of the light ray transform. Our proof also provides a reconstruction algorithm.