Propagation Property and Application to Inverse Scattering for Fractional Powers of Negative Laplacian

East Asian Journal on Applied Mathematics
Vol. 10 No. 1 (2020), pp. 106-122
DOI: 10.4208/eajam.050319.110619
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Author(s)
    Atsuhide Ishida
    Department of Liberal Arts, Faculty of Engineering, Tokyo University of Science, 6-3-1 Niijuku, Katsushika-ku, Tokyo 125-8585, Japan. Tokyo Univ Sci, Fac Engn, Dept Liberal Arts, Katsushika Ku, 6-3-1 Niijuku, Tokyo 1258585, Japan
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1 Department of Liberal Arts, Faculty of Engineering, Tokyo University of Science, 6-3-1 Niijuku, Katsushika-ku, Tokyo 125-8585, Japan.
2 Tokyo Univ Sci, Fac Engn, Dept Liberal Arts, Katsushika Ku, 6-3-1 Niijuku, Tokyo 1258585, Japan
Received
March 5, 2019
Accepted
June 11, 2019
Abstract

The propagation estimate for the usual free Schrödinger operator established by Enss in 1983, was successfully used by Enss and Weder in inverse scattering in 1995. This approach has been called the Enss-Weder time-dependent method. We derive the same type of estimate but for fractional powers of the negative Laplacian and apply it in inverse scattering. It is found that the high-velocity limit of the scattering operator uniquely determines the short-range interactions.

Keywords
Scattering theory inverse problem fractional Laplacian.
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