Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data - Rüland, Angkana; Sincich, Eva (Inverse problems and imaging, 2019)
In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential \begin{document}$ q \in L^{∞}(Ω) $\end{document} in the equation \begin{document}$ ((- Δ)^s+ q)u = 0 in Ω\subset \mathbb{R}^n $\end{document} satisfies the a priori assumption that it is contained in a finite dimensional subspace of \begin{document}$ L^{∞}(Ω) $\end{document} . Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on \begin{document}$ q $\end{document} . We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.