OP-00688 Solution claimed

A fixed-time lower bound for Trotter splitting of the hydrogen ground state

On $L^2(\mathbb{R}^3)$ let $H = -\Delta - |x|^{-1}$, whose normalised ground state $\psi_0(x) = (8\pi)^{-1/2} e^{-|x|/2}$ satisfies $H\psi_0 = -\tfrac14 \psi_0$. For an integer $n \ge 1$ let $$S_n = \bigl(e^{-i|x|^{-1}/n} e^{-i\Delta/n}\bigr)^n$$ be the first-order Trotter splitting of $e^{iH}$ with $n$ steps, where the first factor acts by multiplication and the second is the free Schrödinger group. Becker, Galke, van Luijk and Salzmann proved that for every $\varepsilon > 0$ and every $\psi \in H^2(\mathbb{R}^3)$, $\|S_n\psi - e^{iH}\psi\|_{L^2} = O(n^{-1/4+\varepsilon})$ as $n \to \infty$. Fang and Wu proved that for the ground state of $-\Delta - 2/|x|$ the error of a single step of length $t$ of the first-order Trotter formula is exactly of order $t^{5/4}$ as $t \to 0$.

Problem. Are there constants $c > 0$ and $N$ such that $$\bigl\|S_n\psi_0 - e^{iH}\psi_0\bigr\|_{L^2(\mathbb{R}^3)} \ge c\, n^{-1/4} \qquad (n \ge N)?$$ The final time is fixed at $t = 1$ while the number of steps tends to infinity.

Source: Stated by AIM — Open Applied Problems, after Becker, Galke, van Luijk and Salzmann (2026), who leave an analytical lower bound on the hydrogen ground state open; Fang and Wu (2026) leave the fixed-time many-step lower bound open.

Imported from github.com/MColbrook/AIM.

Classification

Primary: NA — Numerical Analysis
Secondary: MP
Keywords: AIM Open Applied Problems, Coulomb potential, splitting methods, Trotter product formula

References

@article{becker2026,
  author        = {Becker, Simon and Galke, Niklas and van Luijk, Lauritz and Salzmann, Robert},
  title         = {Convergence rates for the {T}rotter splitting for unbounded operators},
  journal       = {Found. Comput. Math.},
  volume        = {26},
  number        = {4},
  year          = {2026},
  pages         = {2471--2524},
  doi           = {10.1007/s10208-025-09730-w},
  eprint        = {2407.04045},
  archivePrefix = {arXiv},
}
@misc{fang2026,
  author        = {Fang, Di and Wu, Xiaoxu},
  title         = {Trotterization with many-body {C}oulomb interactions: convergence for general initial conditions and state-dependent improvements},
  year          = {2026},
  eprint        = {2604.07704},
  archivePrefix = {arXiv},
  note          = {Version 2, 24 August 2026},
}
@misc{aim-hydrogen-trotter-lower-bound,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A fixed-time lower bound for Trotter splitting of the hydrogen ground state}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/561-hydrogen-trotter-lower-bound.md},
  note         = {Entry 561 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Disproof of the proposed fixed-time hydrogen Trotter lower bound v1 GitHub ↗

no authors listed

No: ||S_n psi_0 - e^{iH} psi_0|| <= C n^{-3/8}, so n^{1/4} times the error tends to zero and no eventual lower bound c n^{-1/4} holds..

Disproof · not verified · fidelity not established · posted on GitHub 2 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, 562-hydrogen-trotter-disproof/PROOF.md.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00688, "A fixed-time lower bound for Trotter splitting of the hydrogen ground state", clopen.solutions.