I am interested to know if it is possible to solve generalized Helmholtz equation of the form:
\begin{equation} \nabla^2\psi+k(\bf r)^2\psi=0 \end{equation}
using Fourier transform. I understand that in many cases, $k$ appear as constant, but there may be other cases, where $k$ is a function.
I tried separation of variables in spherical polar coordinates and that looks promising whenever I seek the solution around the ordinary points or the points with removable singularity. But I am interested to know if one can develop a solution just using Fourier transforms. I went through this webpage: http://www.sjsu.edu/faculty/watkins/fouriergenhelm.htm, but the author makes a big jump from constant $k$ to variable $k$ (what he calls $H$) without any explanation.
This brings a relevant question: can we actually write the solution of Schr$\rm{\ddot{o}}$dinger equation with non-zero potential V as Fourier integral? This may be a dumb question and I might be forgetting basics. When we say, $\psi(x)=\int a(k)\ e^{-ikx}\ dk$, do we implicitly assume that $k=\sqrt{\frac{2m}{\hbar^2}(E-V(x))}$?
Any enlightening comment or suggestions are highly appreciated.\
Kolahal