A non-local inverse problem with boundary response

The problem of interest in this article is to study the (non-local) inverse recovering a potential based on boundary measurement associated with fractional Schrödinger equation. Let $0\<a<1$, and let $u$ solve $$ \begin{cases} ((-\Delta)^a + q )u = 0 &\operatorname{in} \Omega,\ \operatorname{supp}, u\subseteq \overline{\Omega}\cup \overline{W} ,\ \cap \overline{\Omega}=\emptyset. \end{cases} We show that, by making an exterior as $\big(u|{W}, \frac{u(x)}{d(x)^a}\big|{\Sigma}\big)$, it possible determine $q$ uniquely $\Omega$, where $\Sigma\subseteq\partial\Omega$ non-empty open subset $d(x)=d(x,\partial\Omega)$ denotes distance function. also discuss local characterization large $a$-harmonic functions ball or half space its applications, which include unique continuation density result.