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

Abstract

We consider the inverse problems of 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. 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 Schrödinger equation.

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• Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem
• Simultaneous recovery of piecewise analytic coefficients in a semilinear elliptic equation
• Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity
• Inverse problem for a nonlocal diffuse optical tomography equation
• The fractional $p$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems
• The Monotonicity Principle for Magnetic Induction Tomography
• Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations
• An Inverse Problem for Non-linear Fractional Magnetic Schrodinger Equation