Simultaneously recovering potentials and embedded obstacles for anisotropic fractional schrödinger operators

Xinlin Cao, Yi Hsuan Lin, Hongyu LIU

Research output: Contribution to journalJournal articlepeer-review

48 Citations (Scopus)


Let A∈Sym(n×n) be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator L s A +q, where (Formula Presented) and q∈L∞. We are concerned with the simultaneous recovery of q and possibly embedded soft or hard obstacles inside q by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain Ω associated with L s A +q. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential q. If multiple measurements are allowed, then the surrounding potential q can also be uniquely recovered. These are surprising findings since in the local case, namely s=1, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

Original languageEnglish
Pages (from-to)197-210
Number of pages14
JournalInverse Problems and Imaging
Issue number1
Publication statusPublished - Feb 2019

Scopus Subject Areas

  • Analysis
  • Modelling and Simulation
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

User-Defined Keywords

  • Fractional elliptic operators
  • Nonlocal inverse problem
  • Runge approximation property
  • Simultaneous recovering
  • Strong uniqueness property


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