Efficiently Solving the Eigenvalue Problem: Sturm-Liouville Equation [URGENT]"

[isison]

[URGENT] Eigenvalue problem

Homework Statement

[PLAIN]http://img228.imageshack.us/img228/4990/111em.png

Homework Equations

Sturm-Liouville equation?

The Attempt at a Solution

I guess I'm just totally lost here. I've no idea how to start. It seems to me that maybe solving for solutions directly is ok, but that's near impossible in this case. I think there's some clever way around.

 

[vela]

Plug f(x) into the eigenvalue equation and use the Laplacian in spherical coordinates in n dimensions to expand the lefthand side.

 

[isison]

Thanks!

How about g(x)? How do I prove g(x) is also a solution?

 

[vela]

Same way, I'd imagine.

 

[isison]

Well, I tried to do the same thing, but I just can't reach the same conclusion. Where does k come from? How do I get that term with k?

Thanks,

 

[vela]

It should come from the Laplacian acting on the spherical harmonic.