Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials

We consider an inverse problem for the fractional Schrödinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calderón problem in a constructive manner. Second, we offer a reconstruction method for unknown obstacles in a given domain. Our method is independent of the dimension and only requires the background solution of the fractional Schrödinger equation.