On the positivity of scattering operators for Poincar\'{e}-Einstein manifolds

In this paper, we mainly study the scattering operators for the Poincar\'{e}-Einstein manifolds. Those operators give the fractional GJMS operators $P_{2\gamma}$ for the conformal infinity. If a Poincar\'{e}-Einstein manifolds $(X^{n+1}, g_+)$ is locally conformally flat and there exists an representative $g$ for the conformal infinity $(M, [g])$ such that the scalar curvature $R$ is a positive constant and $Q_4>0$, then we prove that $P_{2\gamma}$ is positive for $\gamma\in (1,2)$ and thus the first real scattering pole is less than $\frac{n}{2}-2$.