Recovering asymptotics of potentials from the scattering of relativistic Schrödinger operators

We study the stationary scattering for $(-Δ)^{\frac 12} + V(x)$ on $\mathbb{R}^3$. For poly-homogeneous potentials decaying at infinity, we prove that the asymptotics of the potential can be recovered from the scattering matrix at a fixed energy.