Internal control for a non-local Schrödinger equation involving the fractional Laplace operator

We analyze the interior controllability problem for a non-local Schrödinger equation involving the fractional Laplace operator \begin{document}$ (-\Delta)^{\, {s}}{} $\end{document} , \begin{document}$ s\in(0, 1) $\end{document} , on a bounded \begin{document}$ C^{1, 1} $\end{document} domain \begin{document}$ \Omega\subset{\mathbb{R}}^N $\end{document} . We first consider the problem in one space dimension and employ spectral techniques to prove that, for \begin{document}$ s\in[1/2, 1) $\end{document} , null-controllability is achieved through an \begin{document}$ L^2(\omega\times(0, T)) $\end{document} function acting in a subset \begin{document}$ \omega\subset\Omega $\end{document} of the domain. This result is then extended to the multi-dimensional case by applying the classical multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian.