The Calderón problem for the fractional Schrödinger equation

By Tuhin Ghosh, Mikko Salo, Gunther Uhlmann. 2016

175 citations recorded by the source corpus.

We show global uniqueness in an inverse problem for the fractional Schrodinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderon problem.

Subfields: Applied Mathematics, Biomedical Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73cec594fa553c1d675ea3ad198ed8c232647a63

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Citations in the corpus

The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, methodology): “The remaining claims follow from the spectral theorem for G̃μ and from the Fredholm alternative as in [15].”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, methodology): “Unique continuation of the fractional Laplacian has been extensively studied and used to show uniqueness results for fractional Schrödinger equations [14, 27, 28, 75].”
Global uniqueness for the fractional semilinear Schrödinger equation (2017, methodology): “1 generalizes global uniqueness result of [6] for the case where q(x, u) = q(x).”
Fractional anisotropic Calderón problem on closed Riemannian manifolds (2021, background): “The study of the fractional Calderón problem was initiated in [16] where the unknown potential in the fractional Schrödinger equation on a bounded domain in the Euclidean space was determined from exterior measurements.”
Getting Acquainted with the Fractional Laplacian (2017, methodology): “See also [55, 79, 80] for different approaches to approximation methods in fractional settings which lead to new proofs, and very refined and quantitative statements.”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, background): “It intends to generalize the recent study on the Calderón problem for the fractional Schrödinger equation [18].”