quantum mechanics - Does the hydrogen anion have bound excited states?

I'm having some trouble puzzling out the literature regarding the existence of bound excited states in the hydrogen anion, H$^-$.

Wikipedia claims that no such states exist, and that the subject is uncontroversial, stating that

H$^−$ is unusual because, in its free form, it has no bound excited states, as was finally proven in 1977 (Hill 1977)

and citing

1. R.N. Hill, "Proof that the H$^−$ Ion Has Only One Bound State". Phys. Rev. Lett. 38, 643 (1977)

There is a similar further paper by Hill,

2. R.N. Hill, "Proof that the H$^−$ ion has only one bound state. Details and extension to finite nuclear mass", J. Math. Phys. 18, 2316 (1977)

which extends the work to account for the finite mass of the proton.

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On the other hand, upon chasing the highest-cited papers that reference the original ground-state-energy calculation by Bethe [Z. Phys. 57, 815 (1929)], I ran into the review

3. T. Andersen, "Atomic negative ions: structure, dynamics and collisions", Phys. Rep. 394, 157 (2004)

which tells a rather different story. In §4.1, on the hydrogen anion, Andersen states the following:

The H$^−$ ion has two bound states: the ground $\rm 1s^2 \ {}^1S$ state and the doubly excited $\rm 2p^2 \ {}^3P$ state. The latter has not been observed till date. It was predicted computationally nearly 40 years ago and its energy computed repeatedly, most recently and very precise by Bylicki and Bednarz [273]. There is no doubt about its existence, but the experimental non-appearance is linked to the lack of aninitial state from which it canbe reached [273].

Following the kicked can down to Bylicki and Bednarz,

4. M. Bylicki & E. Bednarz, "Nonrelativistic energy of the hydrogen negative ion in the $\rm 2p^2 \ {}^3P^e$ bound state". Phys. Rev. A 67 022503 (2003)

there are further self-assured statements that the state does exist,

The H$^-$ ion has only two bound states: the ground $\rm 1s^2 \ {}^1S$ state and the doubly excited $\rm 2p^2 \ {}^3P$. The former has been investigated both theoretically and experimentally. For references see recent papers of Drake, Cassar, and Nistor [1] and Sims and Hagstrom [2] where the ground-state energy has been computed with an extremely high precision. The other bound state of H$^-$, $\rm 2p^2 \ {}^3P$, has not been observed till date. It was predicted computationally [3] nearly 40 years ago. Its energy was computed repeatedly [4–8] and there is no doubt about its existence. The problem of its experimental nonappearance is due to the lack of an initial state from which it could be reached.

as well as links to a large set of references working on increasing the precision of the theoretical calculation of the energy of this presumed excited state of the system ─ some of them prior to Hill's 1977 work, but also several of them years or decades later from that publication, so they ought to be aware of the theorems in that paper that show that their calculations are impossible. And yet, that reference cluster seems to contain scant or no references to Hill's papers.

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So, onwards to my question: what's going on here?

• Is there an actual conflict in the literature? Or are the two strands of work actually compatible with each other for some reason that I can't see yet?


• Say, do the rigorous theorems of Hill require some additional conditions that can actually be relaxed, and that's what's happening in the numerical calculations?


• Or do the calculations actually end up describing eigenfunctions that are so pathological that they shouldn't be counted as bound states?

There's something funny going on here, but I can't believe that the people writing here were unaware of the other side, so I imagine that there's some aspect of the discussion which is considered as 'obvious' and not mentioned too explicitly, and I'd like to better understand what that is.