(adsbygoogle = window.adsbygoogle || []).push({}); Edit: I have substantially edited this post from its original form, as I realize that it might have fallen under the label of "textbook-style questions".

Really, the heart of my issue here is that, anywhere I look, I can't seem to find a clear description anywhere of the limitations of the separation of variables approach to e.g. Laplace's equation. As in, when exactly are we are allowed to assume that the solution to a differential equation in multiple variables has a form like, for instance,

[tex]\Phi(r,\theta,\phi)=\sum_m\sum_n\sum_kC_{mnk}R_m(r)P_n(\theta)Q_k(\phi)[/tex]

To me, it seems that the validity of this approach depends on whether or not the set of all separable solutions is complete. What I mean is, given a set of elementary separable solutions [itex]\psi_{nmk}=X_n(x)Y_m(y)Z_n(z)[/itex] to a differential equation, it wouldn't be correct to write [itex]\Phi=\sum_n\sum_m\sum_kC_{nmk}\psi_{nmk}[/itex] unless we knew that [itex]\psi_{nmk}[/itex] formed a complete basis for all possible solutions.

Unfortunately, the impression I get from most of my courses is that, since completeness proofs are so hard to come by, we generally just take for granted that certain well-known problems are known to work out.

The subject of my original post --- a worked example of solving for a Green's function in Jackson --- was just a specific instance that I used as an example my frustration. At some point, Jackson makes a conclusion that only appears possible to me if we assume that a function in 4 variables: [itex]r, r' , \theta',[/itex] and [itex]\phi'[/itex] can be composed of solutions which are separable into radial (r,r') and angular parts. As we're dealing with coordinates from two different position vectors, I would hardly think this is a trivial assumption, yet there seems to be no getting around it.

Spoiler: Original post Right now, I'm going through Classical Electrodynamics (Jackson), and am bothered by a particular worked example: (pages 120-122)

Context

We are solving for the Green's function for the region inside a sphere of radius [itex]a[/itex] with Dirichelet boundary conditions. This is to say that we seek a function [itex]G(\mathbf{x},\mathbf{x}')[/itex] such that

[tex]\nabla^2_xG(\mathbf{x},\mathbf{x}') = -4\pi\delta^3(\mathbf{x}-\mathbf{x}')[/tex]

[tex]G(\mathbf{x},\mathbf{x}')|_{\mathbf{x'}\in S}=0[/tex]

(Note: the phrase "with Dirichelet boundary conditions" simply refers to the second equation above, as this constraint allows one to construct the electric potential inside a region in terms of [itex]G[/itex], [itex]\rho[/itex], and the potential at the boundary)

Jackson uses the completeness of the spherical harmonics [itex]Y_{lm}[/itex] to expand both the delta function and the Green's function:

[tex]\delta^3(\mathbf{x}-\mathbf{x}')=\frac{1}{r^2}\delta(r-r')\sum_{l=0}^{\infty}\sum_{m=-l}^l

Y_{lm}^*(\theta',\phi')Y_{lm}(\theta,\phi)[/tex]

[tex]G(\mathbf{x},\mathbf{x}')=\sum_{l=0}^{\infty}\sum_{m=-l}^lA_{lm}(r,r',\theta',\phi')Y_{lm}(\theta,\phi)[/tex]

So far, so good.

My problem

He then states that substitution of these into the first equation above (the one with [itex]\nabla^2_x[/itex]) yields

[tex]A_{lm}(r,r',\theta',\phi')=g_l(r,r')Y^*_{lm}(\theta',\phi')[/tex]

where [itex]g_l(r,r')[/itex] is a function which remains to be solved for.

I find that the substitution produces something like

I'm personally having trouble seeing how this works.

[tex]\sum_{l=0}^{\infty}\sum_{m=-l}^l\left(-\frac{l(l+1)}{r^2}+\frac{1}{r^2}\partial_r(r^2\partial_r)\right)

A_{lm}(r,r',\theta',\phi')Y_{lm}(\theta,\phi)\\

=-4\pi\frac{1}{r^2}\delta(r-r')\sum_{l=0}^{\infty}\sum_{m=-l}^l

Y_{lm}^*(\theta',\phi')Y_{lm}(\theta,\phi)

[/tex]

and from here, I'm not sure what we're allowed to do.

In particular,if we were allowed to assumethat [itex]A_{lm}[/itex] is separable into radial and angular parts:

[tex]A_{lm}(r,r',\theta',\phi')\equiv g_l(r,r')h_{lm}(\theta',\phi')[/tex]

then Jackson's result can by obtained by a simple application of the completeness and orthogonality of [itex]Y_{lm}[/itex]. But I am not convinced that this assumption is valid.Are we allowed to assume separability in this case?

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# Assuming separability when solving for a Green's Function

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