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

TL;DR

This paper investigates an inverse scattering problem for a quantized scalar field governed by the Klein-Gordon equation, demonstrating that the scattering operator uniquely determines the external potential and providing representations for the source components.

Contribution

It establishes the unique determination of the external potential from the scattering operator and derives explicit representations for the source functions in terms of the scattering data.

Findings

The scattering operator uniquely determines the external potential V.

Explicit formulas relate the source functions to the scattering operator.

The analysis applies to a Klein-Gordon equation with external potential and source.

Abstract

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 $\mathscr{S}$.