= Source-position-dependent Fourier Green function
{title2=$G_{\mathbf r'}(\mathbf k;\omega)$}
A static but spatially inhomogeneous linear operator has time <translation> <symmetry> without spatial <translation> <symmetry>. At fixed <source> position $\mathbf r'$, one may still <Fourier transform> its <Green function> in $\mathbf x=\mathbf r-\mathbf r'$. The coefficient retains $\mathbf r'$ as a parameter. A multiplication by $V(\mathbf r)$ becomes a <convolution> with $V_{\mathbf r'}(\mathbf q)=e^{i\mathbf q\cdot\mathbf r'}\widetilde V(\mathbf q)$, not a spatially diagonal multiplier unless $V$ is constant. Suppressing the source-position subscript is only a notational convention.
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