How to prove that there are infinitely many eigenvalues below $1$ of the following differential operator?

Question

The question from Remark (2) below Lemma 7 in Enno Lenzmann: uniqueness of ground states for the pseudorelativistic Hartree equations. 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?

Answer 1

You can find a mathematically precise treatment in On the Energy Levels of the Hydrogen Atom. 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.