SIMULTANEOUSLY RECOVERING POTENTIALS AND EMBEDDED OBSTACLES FOR ANISOTROPIC FRACTIONAL SCHRODINGER OPERATORS

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

11 Scopus Citations
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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)197-210
Journal / PublicationInverse Problems and Imaging
Volume13
Issue number1
Online publishedDec 2018
Publication statusPublished - Feb 2019
Externally publishedYes

Abstract

Let ∈ Sym (n×n) be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator L sA + q, where L sA := (−∇·( (x) ∇))s , s ∈ (0, 1) 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 LsAq. 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 =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.

Research Area(s)

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