Inverse scattering problems for the Hartree equation whose interaction potential decays rapidly

We consider inverse scattering problems for the three-dimensional Hartree equation. We prove that if the unknown interaction potential $V(x)$ of the equation satisfies some rapid decay condition, then we can uniquely determine the exact value of ${\mathrm{\partial }}_{\xi }^{\alpha }\widehat{V}(0)$ for any multi-index $\alpha$ by the knowledge of the scattering operator for the equation. Furthermore, we show some stability estimate for identifying ${\mathrm{\partial }}_{\xi }^{\alpha }\widehat{V}(0)$.