Monotonicity-Based Inversion of the Fractional Schrödinger Equation II. General Potentials and Stability
Bastian Harrach, Yi-Hsuan Lin
Abstract
In this work, we use monotonicity-based methods for the fractional Schrödinger equation with general potentials q in L^\infty(Omega) in a Lipschitz bounded open set Omega \subset R^n in any dimension n in 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 result for the fractional Calderón 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 Schrödinger 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).