Separation of variables for non-central potentials

[dongsh2]

Dear Everyone,

I have a question about the separation of variables for non-central potentials (r, \theta, \phi). In spherical coordinates, such a potential V(r,\theta)=u(r)+f(\theta)/r^2 can be separated. Who knows it could also be separated in other coordinates? Many thanks.

 

[DrDu]

Maybe parabolic co-ordinates. Landau Lifgarbagez, Quantum mechanics, discusses the separability in quite a range of different co-ordinates.

 

[dextercioby]

There's also the famous book by Morse and Feshbach (I don't remember exactly which volume) which discussing the separation of variables in a linear PDE.

 

[dongsh2]

Maybe parabolic co-ordinates. Landau Lifgarbagez, Quantum mechanics, discusses the separability in quite a range of different co-ordinates.

Thanks. But using parabolic ones, how to separate the potential V(r,\theta) to the sum of those variables in parabolic ones.

 

[dongsh2]

There's also the famous book by Morse and Feshbach (I don't remember exactly which volume) which discussing the separation of variables in a linear PDE.

The question is following. For a non-central potential

V(r,\theta)=r^2/2+b/r^2+(c/r^2) [ d/sin^2(\theta) cos^2(\theta) + f/sin^2(\theta)], where b,c,d,f are constants.

We have separated it in spherical coordinates and published. I try to find the possibility in other coordinates.

 

[DrDu]

Maybe you could be more specific about the potential you have in mind. If u(r) and f(theta) are completely general then I don't think that you can find another factorization.
On the other hand there are other non-central potentials which aren't of the form you specified and which are separable.

 

[dongsh2]

This potential was separated and studied last year in spherical coordinates. Which potential (non ours) could be separable in other coordinates? Could you pls tell me? Thanks.

I have sent this question to my friends in USA and France, but I have not received their reply. They are expert in this field.