The purely radial version (without $\theta$ or $\phi$ angles) of the Laplacian is, $\nabla ^2 = \frac{\partial^2}{\partial r^2} + r^{-1} \frac{\partial}{\partial r}$. It has eigenfunctions of the form $A r^{-1}e^{-i(kr)} + B r^{-1}e^{i(kr)}$, which is the sum of a pair of radial waves, where their centers are at the origin of coordinates and $r=\sqrt{x^2 + y^2}$, in cartesian coordinates.
Is $A ||\vec {r}-\vec {r’}||^{-1}e^{-i(k ||\vec {r}-\vec {r’}||)} + B ||\vec {r}-\vec {r’}||^{-1}e^{i(k||\vec {r}-\vec {r’}||)}$, where $\vec {r’}$ is constant and $\vert\lvert \vec {R} \rvert\rvert$ is the magnitude of $\vec {R}$, also an eigenfunction of the above Laplacian, where $r=||\vec {r}||$ in that Laplacian? I think it must be, since this function is the same as the above radial wave eigenfunction, except its center is displaced from the coordinate origin. I just want to be sure.