The Calderón problem for the fractional Schrödinger equation with drift
Cekić, M., Lin, Y.-H., & Rüland, A. (2020). The Calderón problem for the fractional Schrödinger equation with drift. Calculus of Variations and Partial Differential Equations, 59(3), Article 91. https://doi.org/10.1007/s00526-020-01740-6
© The Authors 2020
We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many generic measurements is discussed. Here the genericity is obtained through singularity theory which might also be interesting in the context of hybrid inverse problems. Combined with the results from Ghosh et al. (Uniqueness and reconstruction for the fractional Calderón problem with a single easurement, 2018. arXiv:1801.04449), this yields a finite measurements constructive reconstruction algorithm for the fractional Calderón problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension n≥ 1. ...
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Related funder(s)Academy of Finland
Funding program(s)Academy Project, AoF
Additional information about fundingOpen access funding provided by Projekt DEAL. This project strongly profited from many discussions among the authors during the HIM summer school “Unique continuation and inverse problems” and the MPI MIS summer school “Inverse and Spectral Problems for (Non)-Local Operators” at which the authors participated. The authors would like to thank that Hausdorff Center for Mathematics and the Max-Planck Institute for Mathematics in the Sciences for their support during these two weeks. YHL was supported by the Academy of Finland, under the Project Number 309963, 2018–2019. YHL is now supported by the Ministry of Science and Technology Taiwan, under the Columbus Program: MOST-109-2636-M-009-006, 2020–2025. In the course of writing of this work, MC was supported by the Max-Planck Institute for Mathematics in Bonn. MC is currently supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant Agreement No. 725967). ...
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