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 $\mathit{\varphi }$ obeying a linear Klein-Gordon equation $(\square + m^2 + V) {\bm \phi} = J \mbox{in $\mathbb{R}\times {\mathbb{R}}^3$}$isconsidered,where$V$ is a repulsive external potential and $J$ an external source $J$. We prove that the scattering operator $\mathcal{S}=\mathcal{S}(V,J)$ associated with ${\bm \phi}$uniquelydetermines$V$\mathrm{.}Assumingthat$J$isoftheform$J(t,x)=j(t)\rho(x)$,$(t,x) \in $\mathbb{R}$ \times $\mathbb{R}$^3$,werepresent$\rho$(resp\mathrm{.}$j$)intermsof$j$(resp\mathrm{.}$\rho$)and$\mathscr{S}$.