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\begin{document}

In spherical coordinates, the delta function is \( \frac{1}{r^{2}}\delta (r-r')\delta (cos(\theta )-cos(\theta '))\delta (\phi -\phi ') \). 

The expansion of the angular portion of the delta function is: \( \delta (cos(\theta )-cos(\theta '))\delta (\phi -\phi ')=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}Y_{lm}^{*}(\theta ',\phi ')Y_{lm}(\theta ,\phi ) \).


The laplacian can be expressed as \( \nabla ^{2}=\frac{1}{r}\frac{\partial ^{2}}{\partial r^{2}}r-\frac{1}{r^{2}}L^{i}L^{i} \) where \( L^{i}=\frac{1}{i}\epsilon _{ijk}r^{j}\nabla ^{k} \)

The Green's function will satisfy: \( G(r^{i},r'^{i})=\sum _{l}\sum _{m}A_{lm}(r,r',\theta ',\phi ')Y_{lm}(\theta ,\phi ) \)

Applying the poisson equation gives: \( A_{lm}(r,r',\theta ',\phi ')=A_{l}(r,r')Y_{lm}^{*}(\theta ',\phi ') \) where \( A_{l} \) satisfies: \( \frac{1}{r}\frac{d^{2}}{dr^{2}}rA_{l}-\frac{l(l+1)}{r^{2}}A_{l}=\frac{-4\pi }{r^{2}}\delta (r-r') \). Then
\( A_{l}=A(r^{l}-\frac{a^{2l+1}}{r^{l+1}}) \) for \( a\leq r<r' \) and \( A_{l}=B(\frac{1}{r^{l+1}}-\frac{r^{l}}{b^{2l+1}}) \) for \( r'<r\leq b \).

Some algebra gives \( A=C(r_{min}^{l}-\frac{a^{2l+1}}{r_{min}^{2l+1}})(\frac{1}{r_{max}^{l+1}}-\frac{r_{max}^{l}}{b^{2l+1}}) \)

Integrating the equation for \( A_{l} \), you get: \( \frac{d}{dr}(rA_{l})|_{r'-\epsilon }^{r'+\epsilon }=-4\frac{\pi }{r'} \) This means \( A_{l}(r,r')=C(r_{min}^{l}-\frac{a^{2l+1}}{r_{min}^{l+1}})(\frac{1}{r^{l+1}_{max}}-\frac{r_{max}^{l}}{b^{2l+1}}) \) with \( C=\frac{4\pi }{(2l+1)(1-(\frac{a}{b})^{2l+1})} \).

This means \( G(r^{i},r'^{i})=4\pi \sum _{l,m}\frac{1}{(2l+1)(1-(\frac{a}{b})^{2l+1})} \)

\( *(r_{min}^{l}-\frac{a^{2l+1}}{r_{min}^{l+1}})(\frac{1}{r^{l+1}_{max}}-\frac{r_{max}^{l}}{b^{2l+1}})Y_{lm}^{*}(\Omega ^{'})Y_{lm}(\Omega ) \).



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