Monotonicity-based inversion of the fractional Schr\"odinger equation II. General potentials and stability

In this work, we use monotonicity-based methods for the fractional Schr\"odinger equation with general potentials $q\in L^\infty(\Omega)$ in a Lipschitz bounded open set $\Omega\subset \mathbb R^n$ in any dimension $n\in \mathbb N$. We demonstrate that if-and-only-if monotonicity relations between potentials and the Dirichlet-to-Neumann map hold up to a finite dimensional subspace. Based on these if-and-only-if monotonicity relations, we derive a constructive global uniqueness results for the fractional Calder\'on problem and its linearized version. We also derive a reconstruction method for unknown obstacles in a given domain that only requires the background solution of the fractional Schr\"odinger equation, and we prove uniqueness and Lipschitz stability from finitely many measurements for potentials lying in an a-priori known bounded set in a finite dimensional subset of $L^\infty(\Omega)$.