Hokkaido Mathematical Journal

SASAKI Hironobu, SUZUKI Akito,

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

Hokkaido Mathematical Journal, 40 (2011) pp.149-186

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Abstract

An inverse scattering problem for a quantized scalar field $\phi$ obeying a linear Klein-Gordon equation \[ (☐ + m^2 + V) \phi = J \quad\mbox{in $\mathbb{R} \times \mathbb{R}^3$} \] is considered, where $V$ is a repulsive external potential and $J$ an external source. We prove that the scattering operator $\mathscr{S}= \mathscr{S}(V,J)$ associated with ${\phi}$ 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 $\mathscr{S}$.

MSC(Primary)81T10
MSC(Secondary)81U40, 35R30
Uncontrolled KeywordsQuantum field theory, scattering theory, inverse scattering problem, external field problem