Series-termination quantization of the Coulomb radial equation
= Series-termination quantization of the Coulomb radial equation
For a radial solution regular at the origin, writing $R=e^{-\gamma r}\sum_{n\geq0}a_nr^n$ gives
$$
a_n=\frac{2\gamma n-\beta}{n(n+1)}a_{n-1}.
$$
Normalizability at infinity forces the series to terminate, so $2\gamma N=\beta$ for a positive integer $N$ and $E_N=-mq^4/(2\hbar^2N^2)$.