Source-position-dependent Fourier Green function ($G_{{\mathbf{r}}^{\prime }}(\mathbf{k};\omega )$)

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