Determining a Random Schrödinger Equation with Unknown Source and Potential

Jingzhi Li, Hongyu Liu, Shiqi Ma

Research output: Contribution to journalJournal articlepeer-review

30 Citations (Scopus)
30 Downloads (Pure)


We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrodinger equation with unknown source and potential terms. The well-posedness of the direct scattering problem is first established. Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements. First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns. Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source. Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. The major novelty of our study is that on the one hand, both the random source and the potential are unknown, and on the other hand, both passive and active scattering measurements are used for the recovery in different scenarios.

Original languageEnglish
Pages (from-to)3465-3491
Number of pages27
JournalSIAM Journal on Mathematical Analysis
Issue number4
Publication statusPublished - 21 Aug 2019

Scopus Subject Areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Asymptotic expansion
  • Ergodicity
  • Inverse scattering
  • Passive/active measurements
  • Random Schrodinger equation


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