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 $par
tial_xi^alpha hat{V}(0)$ for any multi-index $alpha$ by the knowledge of the scattering operator for the equation. Furthermore, we show some stability estimate for identifying $partial_xi^alpha hat{V}(0)$.
An inverse scattering problem for a quantized scalar field ${bm phi}$ obeying a linear Klein-Gordon equation $(square + m^2 + V) {bm phi} = J mbox{in $mathbb{R} times mathbb{R}^3$}$ is considered, where $V$ is a repulsive external potential and $J$ a
n external source $J$. We prove that the scattering operator $mathscr{S}= mathscr{S}(V,J)$ associated with ${bm phi}$ uniquely determines $V$. Assuming that $J$ is of the form $J(t,x)=j(t)rho(x)$, $(t,x) in mathbb{R} times mathbb{R}^3$, we represent $rho$ (resp. $j$) in terms of $j$ (resp. $rho$) and $mathscr{S}$.
We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scatt
ering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.
