Determining a Random Schrödinger Operator : Both Potential and Source are Random
Li, J., Liu, H., & Ma, S. (2021). Determining a Random Schrödinger Operator : Both Potential and Source are Random. Communications in Mathematical Physics, 381(2), 527-556. https://doi.org/10.1007/s00220-020-03889-9
Published inCommunications in Mathematical Physics
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
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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Additional information about fundingThe 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.
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