= Solution
Use the physical <Coulomb bound state> boundary condition: the spherically symmetric wavefunction is regular at the origin and square integrable with measure $4\pi r^2dr$. For $E<0$, set
$$
\boxed{K=\frac{\sqrt{-2mE}}{\hbar}>0.}
$$
At large $r$, the decaying branch behaves exponentially as $e^{-Kr}$. Substituting $\psi=f e^{-Kr}$ into the radial <Schrödinger equation> gives
$$
rf''+(2-2Kr)f'+2(\lambda-K)f=0.
$$
The regular solution has a <power series> $f(r)=\sum_{j\geq0}a_jr^j$, with recurrence
$$
\boxed{a_{j+1}=\frac{2[K(j+1)-\lambda]}{(j+1)(j+2)}a_j.}
$$
A nonzero regular solution has $a_0\ne0$. If the series does not terminate, its coefficients eventually have one sign after choosing an overall real phase. For any fixed $0<\delta<1/2$ and sufficiently large $j$, the ratio of consecutive magnitudes is at least $2K(1-\delta)/(j+2)$. The tail therefore grows at least as a positive constant times $e^{2K(1-\delta)r}$ divided by a power of $r$; a finite polynomial of earlier terms cannot cancel it. Multiplication by $e^{-Kr}$ leaves exponential growth, contradicting normalizability.
Thus the series must terminate at degree $N-1$, where $KN=\lambda$ and $N$ is a positive integer. Conversely, termination gives a regular polynomial times $e^{-\lambda r/N}$, which is normalizable. Therefore
$$
\boxed{E_N=-\frac{\hbar^2\lambda^2}{2mN^2}
=-\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2N^2},\qquad N=1,2,\ldots.}
$$
These are the $\ell=0$ members of the hydrogen spectrum. Regularity at the origin is essential: square integrability alone on the punctured radial interval would also admit a singular $1/r$ branch. Such a branch is not a physical eigenfunction of the ordinary three-dimensional Coulomb Hamiltonian; its singularity introduces an inadmissible point-source contribution. In the reduced radial variable $u=r\psi$, the physical condition is $u(0)=0$.
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