A methods preprint reports a mathematically certified range for the ground-state energy of H2+, the hydrogen molecular ion. For the specified non-relativistic Schrödinger operator, it places the lowest energy between −0.5514436010 and −0.5509672618 Ha. The interval is 4.76 × 10−4 Ha wide and includes the accepted reference value, −0.551317.
This is a methods study, not an experiment. Its purpose is to attach a guaranteed lower and upper bound to an eigenvalue calculation—the mathematical search for an energy level. The certificate applies to the stated operator and numerical enclosure; it does not establish that the physical model itself is exact.
How the calculation works
The calculation uses two stages. Stage A supplies a guaranteed bound and a separation certificate; Stage B applies a Lehmann–Goerisch calculation to sharpen the lower bound. The framework then uses explicit Neumann–Dirichlet bracketing—two boundary-condition calculations on a finite box—to bound the matching whole-space eigenvalue, with verified interval arithmetic tracking numerical uncertainty.
The H2+ calculation used a Coulomb operator with nuclei at (−2,0,0) and (2,0,0). Cosine bases were used for the Neumann calculation and sine bases for the Dirichlet calculation. The boxes were nested: half-axes 10 and 8 in the first setup, followed by 20 and 16.
Finite boxes left a domain-truncation gap
On the smaller box, the reported whole-space enclosure was about 1.4 × 10−3 Ha wide. The gap between its Dirichlet and Neumann sides was approximately 1.1 × 10−3 Ha and changed little with spectral order. The paper also describes an irreducible domain-truncation gap of 8.9 × 10−4 Ha for that fixed box.
With the larger box and N=64 per axis, the reported residual width fell to 4.76 × 10−4 Ha; the Dirichlet upper bound was then the limiting side.
A certificate for the next eigenvalue helped tighten the lower edge. A single-vector Lehmann–Goerisch calculation gave a certified lower bound of −0.3395656252 Ha for the second eigenvalue; using that separator improved the lower bound by 5.8 × 10−4 Ha and reduced the enclosure width from 1.05 × 10−3 to 4.76 × 10−4 Ha.
Verified lower bounds stayed within roughly 10−11 of corresponding floating-point values, while interval and floating-point values agreed to 3 × 10−15 at N=48. The agreement is evidence of numerical consistency, not independent validation of the physical model.
A narrower interval came with a caveat
A second route used a globally supported Cartesian-Gaussian trial space, meaning functions defined across the full space. At n=66, it reported a two-sided whole-space interval from −0.551339 to −0.551305 Ha, with width 3.3 × 10−5 Ha. At n=132, the Gaussian upper bound was −0.5513092401 Ha; paired with the certified box lower bound, it gave a reported hybrid width of 1.34 × 10−4 Ha.
Those Gaussian intervals were computed with arbitrary-precision floating point and had not been promoted to formal interval certificates. The box-based interval is formally certified, but its width remains subject to the domain-truncation gap associated with the finite box.
What the certificate covers
The certificate holds the mathematical model fixed. It excludes modeling error, including omitted spin and radiative corrections. It is therefore a proof about the specified operator, not a finding that the non-relativistic model is a complete or experimentally exact description of H2+.
The document is arXiv:2608.25760v1, dated 26 August 2026. The paper says the code and data needed to reproduce the certified bounds are openly available, including Julia pipelines and machine-readable result tables.
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