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I wish to solve an eigenvalue problem:

$$\nabla^{2}f=Ef $$

If I assume spherical symmetry $f(r,\theta,\phi)=f(r)$, I can reduce the problem to 1D:

$$(\frac{2}{r}\frac{d}{dr}+\frac{d^{2}}{dr^{2}})f=Ef$$

My boundary conditions are $f(r=0)=0$ and $f(r=L)=0$, where $L$ is the size of the discrete grid.

I am attempting to solve this problem using a orthogonal basis set expansion, but the issue I am experiencing is the divergence of $r$ at $r=0$. Is there a better method for solving this type of problem? How can I accommodate the singularity?

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  • $\begingroup$ You could multiply both sides of the equation by $r$. Does that work? $\endgroup$ – nicoguaro♦ Nov 20 '17 at 19:35
  • $\begingroup$ @nicoguaro Yes, but then I no longer have an eigenvalue equation. I guess my question is more along the lines of what method is best suited to solve the above problem. $\endgroup$ – Geoffrey Xiao Nov 20 '17 at 19:47
  • $\begingroup$ That's not quite true. You might have something similar to a Sturm-Liouville problem with a weighting function. $\endgroup$ – nicoguaro♦ Nov 20 '17 at 20:15
  • $\begingroup$ @nicoguaro Ok, I see what happens. What would be a simple to implement solution for such a problem? Preferably something other than a shooting method? There seems to be a wide variety of different methods out there. I would like to single out one instead of trying many different types. $\endgroup$ – Geoffrey Xiao Nov 20 '17 at 20:19
  • $\begingroup$ Have you checked these posts: 1, 2? $\endgroup$ – nicoguaro♦ Nov 20 '17 at 21:15

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