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 ≥ 2 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.

In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator (−∇ · (A(x)∇)) s + q), for 0 < s < 1. We determine the unknown bounded potential q from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension n ≥ 2. Our results generalize the recent initiative [17] of introducing and solving inverse problem for fractional Schrödinger operator ((−∆) s + q) for 0 < s < 1. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.

We show global uniqueness in the fractional Calderón problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work [GSU16] considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation result that involve different regularization schemes.This means that zero is not a Dirichlet eigenvalue of (−∆) s + q, and one indeed has a unique solution u ∈ H s (R n ) for any exterior value f . This problem, which was first introduced in [GSU16], should be viewed as a fractional analogue of the classical Calderón problem, which is a well-studied inverse problem for which we refer to the survey article [Uh14] and the references therein. Due to the results of [GSU16], it is known that the Dirichlet-to-Neumann map uniquely determines the potential q, i.e. if q 1 , q 2 ∈ L ∞ (Ω) are such that zero is not a Dirichlet eigenvalue of (−∆) s + q i , i ∈ {1, 2}, thenMoreover, uniqueness holds if the measurements are made on arbitrary, possibly disjoint subsets of the exterior. In [RS17a] this has further been extended to (almost) optimal function spaces, including potentials in L n 2s (Ω). Logarithmic stability for this inverse problem was also proved 2010 Mathematics Subject Classification. 35R11, 35R30, 35J10. Key words and phrases. fractional Calderón problem, single measurement recovery and uniqueness, unique continuation from measurable sets.