A verified calculation has put a certified bracket around the whole-space ground-state energy of H₂⁺. In the main result, the energy lies between -0.5532907722271889 and -0.5507405804827743 in the paper’s energy units, an interval with width 2.550192 × 10−3. The work is presented as an arXiv preprint, version 2, dated 12 October 2026.
The point is not simply to print more digits. The authors aim to make the bounds unconditional by certifying the spectral separator needed by Temple-type and Lehmann-Goerisch inequalities for attractive Coulomb Schrödinger operators. In ordinary language, the separator is a checked barrier around the lowest energy level, part of the chain that turns a numerical estimate into a two-sided bound.
Turning a calculation into a bound
The pipeline starts by replacing unbounded three-dimensional space with a finite box. Neumann boundary values supply the lower side of the box bracket, while Dirichlet values supply the upper side. A confinement check lifts those bounds back to the whole space when the potential floor outside the box lies above the target eigenvalue. The calculation also uses localized Hardy coercivity for the singular part of the operator and a projection stage that certifies the first two eigenvalue bounds, with the second level serving as the separator.
That logic is carried into the numerical calculation with interval matrices and verified eigenvalue routines. The intervals enclose the Ritz, auxiliary and Lehmann-Goerisch quantities instead of leaving them as unchecked point estimates. The reported set contains two H₂⁺ enclosures and three linear three-centre configurations, all produced with an admissible coercivity shift.
A narrow range, and a measurable cost
For the main H₂⁺ run, the box half-length was 10 paper length units, with 64 modes per axis in the production calculation and 80 in the auxiliary one. The resulting interval is deterministic, not a statistical confidence interval: its width describes the verified calculation rather than a probability over repeated samples.
The paper’s conclusion describes the Ω₂ result as a 139 meV width, obtained in 5.3 hours. The authors identify auxiliary resolution as the remaining precision bottleneck.
Nested-box checks helped separate truncation from discretization. For Ω₁, the reported gap was +7.579 × 10−3. For Ω₂, the brackets overlapped, with a difference of −1.002 × 10−3; the analysis reported a true gap of 1.111 × 10−3 against a certified Dirichlet bracket width of 2.113 × 10−3. The practical reading is that, in the larger-box Ω₂ calculation, truncation was below the resolved bracket width. A later remark adds that the Ω₁ result rested on one uncertified parity-sector candidate.
Where the certification stops
One especially sharp result came from a separate global-trial route. Stage A certified a fully even second-level separator at -0.2035 in the paper’s energy units. A global prolate trial then produced a certified Temple enclosure narrower than 7.5 × 10−51 in paper units, equivalent to 1.5 × 10−50 hartree, in about three seconds. That result depends on the special global trial and analytic structure described by the authors, so it is a model-specific demonstration rather than a general promise about runtime or precision.
The linear three-centre test shows where the current certification stops. At nuclear separation 2.8, the narrowest reported interval ran from -0.9365443657837298 to -0.9363042804864575, a width of 2.400853 × 10−4 in paper units. But its lower endpoint was not a certified whole-space bound. The result was explicitly Galerkin-tier, meaning it supports the finite-basis calculation but not the same whole-space certification claimed for the main H₂⁺ run.
The preprint record also contains a precision mismatch. Its abstract reports certified widths of 1.3 × 10−12 and 3.1 × 10−13 for the linear three-centre system at two nuclear separations, while the detailed transferability result reports the wider interval above and does not provide a certified whole-space lower bound. The record does not explain the difference, so the abstract numbers should not be treated as interchangeable with the main H₂⁺ enclosure.
The remaining limits
The scope remains narrow. The current implementation is limited to one-electron systems and does not cover electron-electron interactions. The certified coercivity shift is rigorous but non-sharp by a factor of 1.95, and memory, rather than arithmetic, is the binding resource.
The work reports support from three JSPS KAKENHI grants covering exploratory, scientific research B and scientific research A programmes. The authors report no competing interests and say that code, certificates and a provenance table are archived in the named repository.
Paper data and sources
Original title: From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators
Authors: Xuefeng Liu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text