In order to obtain equation (3), I think I have to do the Fourier transform in the x direction:
\begin{equation}
\tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx
\end{equation}
So I have:
\begin{equation}
-k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...