# Eigenvalues and eigenfunctions of differential operator.

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.

• For your second class the solution can be written in terms of the Parabolic Cylinder Function. – Winther Dec 13 '16 at 3:37
• en.wikipedia.org/wiki/Quantum_harmonic_oscillator – tired Dec 13 '16 at 12:19
• btw, the parameters $a,b$ are useless they might be absorbed in their greek counterparts – tired Dec 13 '16 at 12:23
• @tired that is true, I left it in out of laziness. – Piotr Benedysiuk Dec 13 '16 at 13:08
• The first one is Coulomb potential with angular momentum zero, the second as was said is quantum harmonic oscillator potential. Treatment for both cases is found in any quantum mechanics textbook – Yuriy S Apr 25 '18 at 15:33