### Abstract

In this work, we use monotonicity-based methods for the fractional Schrodinger equation with general potentials q ∈L∞(Ω) in a Lipschitz bounded open set Ω ⊂Rn in any dimension n ∈ 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 Calderon 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 Schrodinger 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∞(Ω).

Original language | English |
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Pages (from-to) | 402-436 |

Number of pages | 35 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 52 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2020 |

### Keywords

- Fractional inverse problem
- Fractional Schrodinger equation
- Lipschitz stability
- Localized potentials
- Loewner order
- Monotonicity