I am trying to interpret the following computation:
Now, for the problem I am working on, which has some subtle differences, my interpretation is as follows:
$$\begin{aligned} \hat{u}_\infty(\omega,\boldsymbol{x},\boldsymbol{\xi})&=\int_{\mathbb{R}^3}V(\boldsymbol{x}')e^{i\omega\boldsymbol{\xi}'\cdot\boldsymbol{x}'/c}e^{-i\omega\boldsymbol{\xi}\cdot\boldsymbol{x}'/c}\,d\boldsymbol{x}'+\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}V(\boldsymbol{x}'_1)e^{-i\omega\boldsymbol{\xi}\cdot\boldsymbol{x}'_1/c} \\ &\cdot\frac{e^{i\omega\|\boldsymbol{x}'_1-\boldsymbol{x}'_1\|/c}}{4\pi\|\boldsymbol{x}'_1-\boldsymbol{x}'_2\|}V(\boldsymbol{x}'_2)e^{i\omega\boldsymbol{\xi}'\cdot\boldsymbol{x}'_2/c}\,d\boldsymbol{x}'_2\,d\boldsymbol{x}'_1+... \\ &...+\int_{\mathbb{R}^3}...\int_{\mathbb{R}^3}V(\boldsymbol{x}'_1)e^{i\omega\boldsymbol{\xi}'\cdot\boldsymbol{x}'/c}\cdot\frac{e^{i\omega\|\boldsymbol{x}'_1-\boldsymbol{x}'_2\|/c}}{4\pi\|\boldsymbol{x}'_1-\boldsymbol{x}'_2\|} \\ &\cdot V(\boldsymbol{x}'_2)\cdot...\cdot V(\boldsymbol{x}'_n)e^{-i\omega\boldsymbol{\xi}\cdot\boldsymbol{x}'_n/c}\,d\boldsymbol{x}'_n...d\boldsymbol{x}'_1+... \end{aligned}$$ Obviously, my $y$ is $x'$ and my $k$ is $\xi$...
Now, I don't understand what they mean by "if Fourier transforms are taken throughout", since the first term in my equation becomes
$$\hat{u}_\infty(\omega,\boldsymbol{x},\boldsymbol{\xi})=\hat{V}(\omega(\boldsymbol{\xi}'-\boldsymbol{\xi})/c)+...$$
which is consistent with their computation (and does not involve any further Fourier transform). However, for the subsequent terms I don't understand what they've done nor how to interpret the $k'''$ et al?
Edit:
If I take Fourier transforms with respect to the $\boldsymbol{x}$ variable, then this gives:
$$\begin{aligned} \mathcal{F}_x\hat{u}_\infty(\omega,\boldsymbol{k},\boldsymbol{\xi})&=\int_{\mathbb{R}^3}\mathcal{F}_x V(\boldsymbol{k}')e^{i\omega\boldsymbol{\xi}'\cdot\boldsymbol{k}'/c}e^{-i\omega\boldsymbol{\xi}\cdot\boldsymbol{k}'/c}\,d\boldsymbol{k}'+\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\mathcal{F}_x V(\boldsymbol{k}'_1)e^{-i\omega\boldsymbol{\xi}\cdot\boldsymbol{k}'_1/c}\frac{e^{i\omega\|\boldsymbol{k}'_1-\boldsymbol{k}'_2\|/c}}{4\pi\|\boldsymbol{k}'_1-\boldsymbol{k}'_2\|} \\&\cdot V(\boldsymbol{k}'_2)e^{i\omega\boldsymbol{\xi}'\cdot\boldsymbol{k}'_2/c}\,d\boldsymbol{k}'_2\,d\boldsymbol{k}'_1+...+\int_{\mathbb{R}^3}...\int_{\mathbb{R}^3}\mathcal{F}_x V(\boldsymbol{k}'_1)e^{i\omega\boldsymbol{\xi}'\cdot\boldsymbol{k}_1/c}\frac{e^{i\omega\|\boldsymbol{k}'_1-\boldsymbol{k}'_2\|/c}}{4\pi\|\boldsymbol{k}'_1-\boldsymbol{k}'_2\|}\\&\cdot V(\boldsymbol{k}'_2)\cdot...\cdot V(\boldsymbol{k}'_n)e^{-i\omega\boldsymbol{\xi}\cdot\boldsymbol{k}'_n/c}\,d\boldsymbol{k}'_n...d\boldsymbol{k}'_1+...\end{aligned}$$
Again, I can interpret the first addendum as $\mathcal{F}_x\hat{V}(\omega(\boldsymbol{\xi}'-\boldsymbol{\xi}/c)$, but I can't seem to equate the other addendums to something similar to what they have in the article...
The second term is $$\int V(k'-k'') \frac{1}{k''^2 - k^2}V(k'' - k) dk''$$
$$=\int \int e^{i(k'-k'')x}V(x)dx \frac{1}{k''^2 - k^2} \int e^{i(k''-k)y}V(y)dy dk''$$
$$=\int \int e^{ik'x} V(x) \left[\int e^{-ik''x} \frac{1}{k''^2 - k^2} e^{ik''y} dk''\right] e^{-iky}V(y)dx dy$$
$$=\int \int e^{ik'x} V(x) \left[-2 \pi^2\frac{e^{i|k||x-y|}}{|x-y|}\right] e^{-iky}V(y)dx dy$$
The last step is in Max Kreuzer's Lecture Notes equations 8.107 and 8.115 .