Vol. 13, No. 2, 2020

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The Calderón problem for the fractional Schrödinger equation

Tuhin Ghosh, Mikko Salo and Gunther Uhlmann

Vol. 13 (2020), No. 2, 455–475
Abstract

We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension 1 and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderón problem.

Keywords
inverse problem, Calderón problem, fractional Laplacian, approximation property
Mathematical Subject Classification 2010
Primary: 26A33, 35J10, 35R30
Secondary: 35J70
Milestones
Received: 9 July 2018
Revised: 8 November 2018
Accepted: 23 February 2019
Published: 19 March 2020
Authors
Tuhin Ghosh
Jockey Club Institute for Advanced Study
Hong Kong University of Science and Technology
Kowloon
Hong Kong
Mikko Salo
Department of Mathematics and Statistics
University of Jyväskylä
Jyväskylä
Finland
Gunther Uhlmann
Department of Mathematics
University of Washington
Seattle, WA
United States
Jockey Club Institute for Advanced Study
Hong Kong University of Science and Technology
Kowloon
Hong Kong