Result 262, Mathematical physics

Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2<γ<3/21/2\lt \gamma\lt 3/2 and arbitrary finite-matrix potentials W ≥ 0 with ∫tr(Wγ+1/2)<∞\int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty: the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech2 solitons with independent scales and centers, and zero channels.

Lean formalization Proof

The bigger picture

Why it matters

How strongly can an attractive quantum system bind particles when it has several interacting internal channels? The manuscripts report a sharp bound that does not worsen as channels are added, and identify every potential attaining it.

What changes?

For one-dimensional systems with any finite number of channels, the unreviewed manuscripts bound the sum of all absolute negative eigenvalues raised to gamma, for gamma strictly between one-half and three-halves. The bound is the size-independent scalar one-bound-state constant times the integral of the trace of the matrix power W to gamma plus one-half. Here W is a measurable Hermitian positive semidefinite matrix describing attraction; that integral must be finite. Arbitrary ranks and matrices that do not commute across positions are allowed.

What does that help mathematicians do?

The equality classification identifies exactly which potentials exhaust this spectral bound. In one fixed unitary basis, every such potential must split into independent scalar channels with squared hyperbolic secant profiles, each supporting one bound state, plus zero channels. Their scales and centers can vary independently. Researchers could therefore rule out exact saturation for any potential that cannot be diagonalized in a single fixed basis, even though the inequality allows such potentials.

Are there practical applications?

The immediate value is foundational in mathematical physics: the claimed estimate controls a weighted total of bound-state energies using an integral of the attractive potential, without a penalty for additional channels. Its equality cases provide exact reference profiles for spectral analysis. These are mathematical consequences, not demonstrated computational improvements or experimental applications.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality

October 5, 2026 28 pages

We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and 1/2<γ<3/21/2\lt \gamma\lt 3/2. For measurable Hermitian positive semidefinite potentials W with ∫Rtr(Wγ+1/2)<∞\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty, equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026,
  author = {{OpenAI}},
  title = {{Equality cases in the sharp one-dimensional matrix Lieb--Thirring inequality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf}{OAI:Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026}},
  year = {2026}
}

Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials

October 5, 2026 24 pages

We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent 1/2<γ<3/21/2\lt \gamma\lt 3/2. The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying ∫Rtr(Wγ+1/2)<∞\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty. The matrices may have arbitrary rank and need not commute at different points.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026,
  author = {{OpenAI}},
  title = {{Sharp one-dimensional Lieb--Thirring inequalities for matrix potentials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf}{OAI:Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026}},
  year = {2026}
}

Sharp one-dimensional Lieb–Thirring constants

September 23, 2026 21 pages Main result formalized in Lean

We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every 12<γ<32\frac12\lt \gamma\lt \frac32, the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in Lγ+1/2(R)L^{\gamma+1/2}(\mathbb R), with all negative eigenvalues included.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026,
  author = {{OpenAI}},
  title = {{Sharp one-dimensional Lieb--Thirring constants}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf}{OAI:Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/262.md.

Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result determines the sharp one-dimensional Lieb–Thirring constant for 1/2<γ<3/21/2<\gamma<3/2. For every nonnegative W∈Lγ+1/2(R)W\in L^{\gamma+1/2}(\mathbb R), it bounds the full negative-eigenvalue moment of −d2/dx2−W-d^2/dx^2-W by the one-bound-state constant times the potential integral. This constant is optimal and is attained by (r+1)sech2(rx)(r+1)\mathrm{sech}^2(rx), where r=(γ−1/2)−1r=(\gamma-1/2)^{-1}. No finite spectral cutoff is imposed.

Comparator links

Result Comparator statement
Sharp one-dimensional Lieb–Thirring inequality LiebThirring.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.