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Determining a Random Schrödinger Operator: Both Potential and Source are Random

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Abstract

We study an inverse scattering problem associated with a Schrödinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We then derive two unique recovery results in determining the rough strengths of the random source and the random potential, by using the corresponding far-field data. The first recovery result shows that a single realization of the passive scattering measurements uniquely recovers the rough strength of the random source. The second one shows that, by a single realization of the backscattering data, the rough strength of the random potential can be recovered. The ergodicity is used to establish the single realization recovery. The asymptotic arguments in our study are based on techniques from the theory of pseudodifferential operators and microlocal analysis.

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Acknowledgements

The work of J. Li was partially supported by the NSF of China under the Grant Nos. 11571161 and 11731006, the Shenzhen Sci-Tech Fund No. JCYJ20170818153840322. The work of H. Liu was partially supported by Hong Kong RGC general research funds, Nos. 12302017, 12301218, 12302919 and 12301420. The authors would like to thank the anonymous referee for many insightful and constructive comments and suggestions, which have led to significant improvements on the results as well as the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Jingzhi Li

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

    Hongyu Liu

  3. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Shiqi Ma

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  1. Jingzhi Li
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  2. Hongyu Liu
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  3. Shiqi Ma
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Correspondence to Hongyu Liu.

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Communicated by M. Hairer

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Li, J., Liu, H. & Ma, S. Determining a Random Schrödinger Operator: Both Potential and Source are Random. Commun. Math. Phys. 381, 527–556 (2021). https://doi.org/10.1007/s00220-020-03889-9

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  • DOI: https://doi.org/10.1007/s00220-020-03889-9

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