I'm trying to find the eigenvalues (atleast the lowest) and eigenvectors of
$$\alpha \frac{\partial^2}{\partial r^2} + \beta V(r) $$
with $\alpha$ and $\beta$ constant, $V(r) = \frac{a}{r}$ and for $V(r) = br^2$.
In case of the first potential, one solution I have found is of the form $r \exp(-\lambda r)$ but it's only one, and I expect a whole range of possible solutions.
On the second potential I'm simply failing to see a solution.
Can some one give me the general direction I should move to? Straight up solutions are nice as well.