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Inverse problem for a random Schrödinger equation with unknown source and potential

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Abstract

In this paper, we study an inverse scattering problem associated with the time-harmonic Schrödinger equation where both the potential and the source terms are unknown. The source term is assumed to be a generalised Gaussian random distribution of the microlocally isotropic type, whereas the potential function is assumed to be deterministic. The well-posedness of the forward scattering problem is first established in a proper sense. It is then proved that the rough strength of the random source can be uniquely recovered, independent of the unknown potential, by a single realisation of the passive scattering measurement. In addition to the use of a single sample of the passive measurement for two unknowns, another significant feature of our result is that there is no geometric restriction on the supports of the source and the potential: they can be separated, or overlapped, or one containing the other.

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Acknowledgements

The research of H Liu was supported by NSFC/RGC Joint Research Scheme, N_CityU101/21, ANR/RGC Joint Research Scheme, A-CityU203/19, and the Hong Kong RGC General Research Funds (projects 11311122, 12301420 and 11300821).

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China

    Hongyu Liu

  2. School of Mathematics, Jilin University, Changchun, China

    Shiqi Ma

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  1. Hongyu Liu
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Correspondence to Hongyu Liu.

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Liu, H., Ma, S. Inverse problem for a random Schrödinger equation with unknown source and potential. Math. Z. 304, 28 (2023). https://doi.org/10.1007/s00209-023-03289-4

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  • DOI: https://doi.org/10.1007/s00209-023-03289-4

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