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On the positivity of scattering operators for Poincaré-Einstein manifolds

In this paper, we mainly study the scattering operators for the Poincaré-Einstein manifolds. Those operators give the fractional GJMS operators $P_{2γ}$ for the conformal infinity. If a Poincaré-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 $P_{2γ}$ is positive for $γ\in (1,2)$ and thus the first real scattering pole is less than $\frac{n}{2}-2$.