VibeMathedMath problems solved with AI
Sign in

The one-dimensional Lieb-Thirring conjecture for 1/2<γ<3/21/2<\gamma<3/2

For 0≤W∈Lγ+1/2(R)0\le W\in L^{\gamma+1/2}(\mathbb R) the Lieb-Thirring inequality reads ∑j∣λj(−d2dx2−W)∣γ≤Lγ,1∫RWγ+1/2\sum_j|\lambda_j(-\frac{d^2}{dx^2}-W)|^\gamma\le L_{\gamma,1}\int_{\mathbb R}W^{\gamma+1/2}. The sharp constant is the semiclassical one for γ≥3/2\gamma\ge3/2 (Lieb-Thirring, Aizenman-Lieb) and equals 1/21/2 at γ=1/2\gamma=1/2 (Hundertmark-Lieb-Thomas). For 1/2<γ<3/21/2<\gamma<3/2, Lieb and Thirring conjectured that the optimal constant equals the one-bound-state constant Lγ,1(1)L^{(1)}_{\gamma,1}, the best constant when only the lowest eigenvalue is counted. Is Lγ,1=Lγ,1(1)L_{\gamma,1}=L^{(1)}_{\gamma,1} for every 1/2<γ<3/21/2<\gamma<3/2?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Spectral theory; sharp Lieb-Thirring constants
Posed by
Elliott H. Lieb and Walter E. Thirring (1976)
Year posed
1976
Years open
50y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
36 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every 1/2<γ<3/21/2<\gamma<3/2 and 0≤W∈Lγ+1/2(R)0\le W\in L^{\gamma+1/2}(\mathbb R), Tr(H−W)−γ≤2(γ−1/2γ+1/2)γ−1/2Lγ,1cl∫Wγ+1/2\mathrm{Tr}(H_{-W})_-^\gamma\le2(\frac{\gamma-1/2}{\gamma+1/2})^{\gamma-1/2}L^{cl}_{\gamma,1}\int W^{\gamma+1/2}; the constant equals Lγ,1(1)L^{(1)}_{\gamma,1} and is attained by (r+1)sech2(rx)(r+1)\mathrm{sech}^2(rx), r=(γ−1/2)−1r=(\gamma-1/2)^{-1}. Companions extend the same constant to Hermitian matrix potentials of any finite size (noncommuting, any rank) and classify all equality cases as direct sums of scalar solitons and zero channels. The scalar paper does not classify optimizers, and nothing is claimed in dimensions above one.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). All three manuscripts are authored 'OpenAI' and name no human author. The matrix-potential paper and the equality-case paper (both dated 2026-10-05) build on the scalar paper's action and continuation framework.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the scalar paper were read against the conjecture. formalization.yaml lists ComparatorChallenges/LiebThirring.json with declaration OAI.SharpLiebThirring.sharp_lieb_thirring (file OAI/Analysis/LiebThirring/Main.lean, present at the pinned commit). The statement was read here: for every 1/2 < gamma < 3/2 and every a.e. nonnegative W in L^(gamma+1/2), the supremum over finite orthonormal families of weak negative-energy eigenfunctions of the sum of k_i^(2 gamma) is at most the explicit constant times the potential integral; the optimal constant and the one-bound-state constant both equal that value; and the sech^2 potential attains it. This is the headline claim. Not rebuilt here. The matrix-potential and equality-case companions are not formalized.

Sources

Changelog1 change

Discussion