Uniqueness of the static spacetimes with a photon sphere in Einstein-scalar field theory

In the present paper we prove a uniqueness theorem for the static and asymptotically flat solutions to the Einstein-scalar field equations which possess a photon sphere. We show that such solutions are uniquely specified by their mass M and scalar charge q and that they are isometric to the Janis-Newman-Winicour solution with the same mass and scalar charge subject to the inequality q(2)/M-2 < 3.