= Solution
<Separation of variables> makes the angular factor a <spherical harmonic>. Applying the <Quantum Hamiltonian> gives
$$
\hat H(RY_{lm})=
Y_{lm}\left[-\frac{1}{2r^2}\frac{d}{dr}\left(r^2R'\right)
+\frac{l(l+1)}{2r^2}R-\frac{R}{r}\right].
$$
Consequently $RY_{lm}$ is an <energy eigenfunction> \b[provided the radial factor satisfies]
$$
\boxed{-\frac{1}{2r^2}\frac{d}{dr}(r^2R')+
\left[\frac{l(l+1)}{2r^2}-\frac1r\right]R=ER.}
$$
This is the <radial Schrödinger equation for a central potential>, here with the attractive <Coulomb potential>. An arbitrary $R$ does not give an <eigenfunction>; regularity, decay and <wavefunction normalization> select admissible bound <quantum states>.
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