An inverse scattering problem for the Klein-Gordon equation with a classical source in quantum field theory

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 $J$ an 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 $mathscr{S}$.