I once did some work on using orthogonal function expansions for fitting 3D distribution functions. To ensure completeness over $L^2$ (which was considered sufficient even though technically a distribution function is an element of $L^1$), we used eigenfunctions of a self-adjoint operator. Initially we used the bound states of the Schrodinger hydrogen atom, but ran into a problem: they didn't seem to be complete over $L^2$. After a few tries to fix it we gave up and switched to the isotropic quantum harmonic oscillator, which seemed to do the trick.


At the time we believed the lack of completeness was because of the existence of free particle solutions to the hydrogen atom, which the QHO does not have. However, I've recently revisited the problem and am unsure this is true. From what I can tell, there are three possibilities


1. We were correct in our assessment of what was wrong. In the case of the hydrogen atom basis, completeness would require a both a sum over the bound states and some sort of integral transform over the free states.
2. We were incorrect to assess that it was wrong. The hydrogen atom bound states are complete and our implementation was incorrect, with the error being quietly fixed when we switched the code to use QHO functions.
3. We were correct that we were wrong, but not why. The hydrogen atom bound states are not complete over $L^2$, but it's for some other reasons involving subtleties (or not-so-subtleties) that we missed.


So the question is, which (if any) of these is correct and, relatedly, what are the conditions for the eigenfunctions of a self-adjoint operator to be complete over $L^2$?



Completeness of the solutions to the Schrödinger Hydrogen Atom
The attractive Coulomb potential has a bound spectrum for $E<0$ and a continuous spectrum for $E>0$, and if you consider the full spectrum you obtain a complete set of eigenfunctions, as proven for example in [Direct demonstration of the completeness of the eigenstates of the Schrödinger equation with local and non-local potentials bearing a Coulomb tail](https://arxiv.org/abs/0712.2651) (2007).


