Coulomb radial origin condition ($u=rR,u(0)=0,R(r)\sim r^{\ell }$)

For the usual three-dimensional Coulomb potential Hamiltonian operator, a separated radial wavefunction $R(r)$ uses the reduced function $u=rR$, whose radial norm has measure $dr$. The regular physical operator domain imposes $u(0)=0$. Near the origin the leading radial equation gives the indicial equation $c(c-1)-\ell (\ell +1)=0$ for $u\sim r^c$, with roots $c=\ell +1$ and $c=-\ell$.

The regular root gives $R\sim r^{\ell }$. The other root gives $R\sim r^{-\ell -1}$, and its radial probability integral ${\int }_0^{\epsilon }{r}^2|R{|}^{2}dr\sim {\int }_0^{\epsilon }{r}^{-2\ell }dr$ diverges for $\ell \ge 1$. At $\ell =0$, $R\sim 1/r$ is locally square-integrable in three dimensions, but violates $u(0)=0$ and has divergent kinetic energy integral ${\int }_0^{\epsilon }{r}^2|R^{\prime }{|}^{2}dr\sim {\int }_0^{\epsilon }{r}^{-2}dr$. Thus local wavefunction normalization alone is insufficient to select the physical branch in this exceptional case; the standard Coulomb energy domain supplies the missing condition.