# Author correction: Incompatible Coulomb hamiltonian extensions

G. Abramovici

► To cite this version:

G. Abramovici. Author correction: Incompatible Coulomb hamiltonian extensions. Scientific Reports, 2020, 10, 10.1038/s41598-020-62144-2. hal-05092332

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Submitted on 2 Jun 2025

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# Author Correction: Incompatible Coulomb hamiltonian extensions

G. Abramovici

Correction to: *Scientific Reports* https://doi.org/10.1038/s41598-020-62144-2, published online 29 April 2020

Published online: 02 July 2020

This Article contains errors in Figures 1 and 4. In Figure 1, some of the curves are incorrect due to a change in the behaviour of the function « HarmonicNumber » in Mathematica since the figure was created in 2015. In Figure 4, the curves have not been normalised as stated in the legend, leading to scaling change in the y-axis. The correct Figures 1 and 4 appear below as Figures 1 and 2 respectively.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Here are the curves $\eta \mapsto \langle\varphi_{-\frac{\eta}{2n}}|\varphi_{-\frac{\eta}{2n}}\rangle$, for $n = -1/2$ (dashed line), $n = -3/4$ (dotted line), $n = -1$ (plain line), $n = -5/4$ (dot-dashed line), $n = -3/2$ (dashed line), $n = -7/4$ (dotted line) and $n = -2$ (plain line). The zeros of each curve read $\eta = \frac{\frac{\eta}{2}}{2\sqrt{-e}}$ where $e \in \mathcal{S}_\omega$, with $\omega = -g_b(n)$, as explained further on. The curves seem to form pairs corresponding to $(n, n+1)$, in particular, one could believe that each pair intersects on the $\eta$-axis (abscissa), but this is wrong, except for $(n, n+1) = (-2, -1)$ which correspond to the same Rydberg set $\mathcal{S}_\infty$. All the other intersections are only close to zero, so that, indeed, $\mathcal{S}_{-g_b(n)} \neq \mathcal{S}_{-g_b(n+1)}$. $\eta = n$ is missing, because $\langle\varphi_{-\frac{\eta}{2n}}|\varphi_{-\frac{\eta}{2n}}\rangle \neq 0$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Here is a zoom of Fig. 1 in the interval $[-\frac{1}{2}, 0]$.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

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