Monotonicity-based inversion of the fractional Schr\"odinger equation I. Positive potentials

We consider the inverse problems of for the fractional Schr"odinger 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 Calder'on problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\ge 2$ and only requires the background solution of the fractional Schr"odinger equation.