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A Coulomb bound state is a physical normalizable eigenfunction for an attractive potential $V(r)=-\gamma /r$, $\gamma >0$. For the usual regular boundary condition at the origin its energies are the displayed values, $N=1,2,\dots$. The spherically symmetric states have orbital angular momentum zero; series-termination quantization of the Coulomb radial equation constructs them as a polynomial times a decaying exponential. Square integrability on the punctured radial interval alone does not replace the physical condition at the origin.

Put $R=f{e}^{-\gamma r}$. ThenAfter substitution and cancellation of the ${\gamma }^{2}f$ terms, the equation becomesFor the power series $f={\sum }_{n=0}^{\infty }{a}_n{r}^n$, the coefficient of $r^{n-1}$ isThus

If the series does not terminate, then $a_n/a_{n-1}\sim 2\gamma /n$, so $f$ acquires the large-$r$ behaviour $e^{2\gamma r}$ and $R$ grows like $e^{\gamma r}$. Normalizability therefore requires termination, which occurs whenfor some positive integer $N$. HenceThe ground state has $N=1$, so the hydrogen ground-state energy isThis is the series-termination quantization of the Coulomb radial equation.