I am trying to find a solution to $$\displaystyle \left[-\frac{1}{2}\nabla^2 - \frac{2}{r} + C(r)\right]\phi(r) = E\phi(r)$$
where C(r) is a known function of r. I am just looking for some help on the strategy to solve explicitly for ϕ(r). I have read through the solution for hydrogen using separation of variables but
1) I don't really understand it, they make a lot of handwavy arguments about terms disappearing :)
2) How does one handle the extra $C(r)$?