Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets

Abstract We consider a restricted Dirichlet-to-Neumann map Λ S , R T associated with the operator ∂ t 2 − Δ g + A + q where Δ g is the Laplace-Beltrami operator of a Riemannian manifold ( M , g ) , and A and q are a vector field and a function on M . The restriction Λ S , R T corresponds to the case where the Dirichlet traces are supported on ( 0 , T ) × S and the Neumann traces are restricted on ( 0 , T ) × R . Here S and R are open sets, which may be disjoint, on the boundary of M . We show that Λ S , R T determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T / 2 . We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.