The question from Remark (2) below Lemma 7 in [Enno Lenzmann: uniqueness of ground states for the pseudorelativistic Hartree equations](https://msp.org/apde/2009/2-1/p01.xhtml). Let $l \ge 1$, and consider the following operator
\begin{equation}
H=-\partial\_{rr} - \frac{2}{r} \partial\_r + 1 + \frac{l(l+1)}{r^2} - \frac{\alpha}{r}
\end{equation}
on $L^2(\mathbb{R}^+,r^2 dr)$, where $\alpha>0$ is a fixed constant. Then how can we check there are infinitely many eigenvalues below $1$ of $H$? Lenzamnn said that we can use the properties of hydrogen atom Hamiltonian to get it, although I have searched the Internet for some informations, many of the them are from the point of physics, and I wonder how to check it precisely? Or can someone give me some useful materials about it?



How to prove that there are infinitely many eigenvalues below $1$ of the following differential operator?
You can find a mathematically precise treatment in [On the Energy Levels of the Hydrogen Atom](https://cds.cern.ch/record/251291/files/9305102.pdf). For $l=0$ the eigenvalues $1-\kappa^2$ of $H$ are given by the square integrable solutions of
$$\left(-\frac{d^2}{dr^2}-\frac{\alpha}{r}\right)v(r)=-\kappa^2v(r).$$
The solutions are Whittaker functions and $\kappa=\tfrac{1}{2}\alpha/n$, $n=1,2,\ldots$.


For $l\geq 1$ the requirement that the eigenfunctions of $H$ are square integrable at the origin restricts $n$ to values $>l$, but there are still infinitely many of them.


