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Determine a Magnetic Schrödinger Operator with a Bounded Magnetic Potential from Partial Data in a Slab

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Abstract

We study an inverse boundary value problem with partial data in an infinite slab in \(\mathbb {R}^{n}\), \(n\ge 3\), for the magnetic Schrödinger operator with bounded magnetic potential and electric potential. We show that the magnetic field and the electric potential can be uniquely determined, when the Dirichlet and Neumann data are given on either different boundary hyperplanes or on the same boundary hyperplanes of the slab. These generalize the results in Krupchyk et al. (Commun Math Phys 312:87–126, 2012), where the same uniqueness results were established when the magnetic potential is Lipschitz continuous. The proof is based on the complex geometric optics solutions constructed in Krupchyk and Uhlmann (Commun Math Phys 327:993–1009, 2014), which are special solutions to the magnetic Schrödinger equation with \(L^{\infty }\) magnetic and electric potentials in a bounded domain.

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Acknowledgements

The authors would like to thank Prof. Katya Krupchyk for pointing out an error in the previous draft. The research of the first author was partly supported by a Clemson Support for Early Exploration and Development (CU SEED) Grant. The research of the second author was partly supported by NSF Grant DMS-1715178, an AMS-Simons travel Grant, and a start-up fund from Michigan State University. The authors would also like to thank the referee for many useful suggestions that result a better presentation of the paper.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634, USA

    Shitao Liu

  2. Department of Computational Mathematics, Engineering and Sciences, Michigan State University, East Lansing, MI, 48824, USA

    Yang Yang

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  1. Shitao Liu
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  2. Yang Yang
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Correspondence to Shitao Liu.

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Liu, S., Yang, Y. Determine a Magnetic Schrödinger Operator with a Bounded Magnetic Potential from Partial Data in a Slab. Appl Math Optim 83, 277–296 (2021). https://doi.org/10.1007/s00245-018-9537-2

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