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 $\partial_ξ^α\hat{V}(0)$ for any multi-index $α$ by the knowledge of the scattering operator for the equation. Furthermore, we show some stability estimate for identifying $\partial_ξ^α\hat{V}(0)$.

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