Recovering asymptotics of potentials from the scattering of relativistic Schrödinger operators

We study the stationary scattering for $(-\mathrm{\Delta }{)}^{\frac{1}{2}}+V(x)$ 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.