Prequel: I am trying to develop a solution for a scattering the problem with boundary conditions on a torus. Suitably enough there are toroidal coordinates. Unfortunatelly Helmholtz equation is not separable in these coordinates, but Laplace equation is. My question is therefore as follows.
Question: Given a family of solutions to Laplace equation (${\phi_n}$):
$\nabla^2 \phi_n\left(\vec{r}\right) =0$
Is there a general way to construct a solution to Helmholtz equation out of them? i.e. find $F=F\left(\vec{r}\right)$ s.t.
$\left(\nabla^2+k^2\right)F=0,\quad k \in \mathbb{R}$
For example, if $\phi=1/r$ it is well-known that $F=\exp\left(\pm i k/\phi\right)\phi=\frac{\exp\left(\pm i k r\right)}{r}$ is a solution to Helmholtz equation (for $r\neq 0$).
I will appreciate any pointers on this matter!
PS: I understand, of course, that the solution to Helmholtz equation will still not be separable in toroidal coordinates.
Thank you