2026/07/24 by Gunter Uhlmann, Philipp Zimmermann
#math.AP
We study an inverse obstacle problem for the fractional Schrödinger operator (-Δ)s+q, 0<s<1. For each exterior datum, the state is constrained by a prescribed obstacle in a bounded domain and satisfies the fractional Schrödinger equation only in the associated noncontact set. This set is unknown and depends on the coefficient, so the exterior Dirichlet-to-Neumann map is nonlinear. We show that the nonlocal character of the equation gives a direct way around this moving-free-boundary difficulty. Equality of one obstacle measurement on an exterior open set forces equality of the two corresponding obstacle states in the whole space. In their common noncontact set one then obtains (q1-q2)u=0. This identity yields recovery of the potential in the exposed region by measurable unique continuation when s∈[1/4,1), and by the usual unique continuation principle together with continuity when the potentials are continuous. We also prove a geometric coverage theorem for nonnegative potentials: rational positive scalings of one nontrivial nonnegative exterior datum expose the whole domain up to a null set. Consequently, under the corresponding assumptions, a countable family of nonlinear exterior obstacle measurements determines the potential globally.