The radial Schrodinger equation for H-like atom
- $$\left( { - \frac{1}{{2r}}\frac{{{d^2}}}{{d{r^2}}}r + \frac{{l\left( {l + 1} \right)}}{{2{r^2}}} - \frac{Z}{r}} \right){\Psi _{nl}}\left( r \right) = {E_{nl}}{\Psi _{nl}}\left( r \right)$$
where $l$ is the angular quantum number.
The Green's function for Eq. (1) [1]
- $${G_l}\left( {r,r',E} \right) = {{2i} \over {\sqrt {rr'} }}{\left( { - 1} \right)^{l + 1}}{\int\limits_1^{ + \infty } {{1 \over {\sqrt {{\xi ^2} - 1} }}\left( {{{\xi + 1} \over {\xi - 1}}} \right)} ^{i{1 \over {\sqrt {2E} }}}}{J_{2l + 1}}\left( {2\sqrt {2E} \sqrt {rr'} \sqrt {{\xi ^2} - 1} } \right){e^{i\sqrt {2E} \xi \left( {r + r'} \right)}}d\xi $$
where ${J_\alpha }\left( z \right)$ is the Bessel function.
[1] Zon, B.A.; Manakov, N.L.; Rapoport, L.P. Theory of manyphotons processes in atoms. Moscow: Atomizdat, 1978