The operator \begin{align*} H_C=-\frac{\hbar^2}{2\mu}\Delta-\frac{\kappa}{|x|} \end{align*} with initial domain $C_c^\infty(\mathbb R^3)$ is essentially self-adjoint on $L^2(\mathbb R^3)$. Its closure is bounded below, has operator domain $H^2(\mathbb R^3)$ under the Kato-Rellich realisation, and is the physical hydrogen Hamiltonian without spin.