Lipschitz stability for the Finite Dimensional Fractional Calderón Problem with Finite Cauchy Data

Abstract: In this note we discuss the conditional stability issue for the finite dimensional Calder\'on problem for the fractional Schr\"{o}dinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $q \in L^{{\infty}(\Omega)} $ in the equation $((-\Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n$ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $L^{{\infty}(\Omega)$.} Under this condition we prove Lipschitz stability estimates for the fractional Calder\'on problem by means of finitely many Cauchy data depending on $q$. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schr\"odinger equation. Our result relies on the strong Runge approximation property of the fractional Schr\"odinger equation.