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Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials
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Abstract
We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent $1/2\lt \gamma\lt 3/2$. 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 $\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$. The matrices may have arbitrary rank and need not commute at different points.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Sharp finite-matrix Lieb–Thirring inequalities and all equality cases: Lean scope notes (
lean/docs/262.md, OpenAI Math Release, 2026-10-06). The documented formalization covers only the September scalar companion's inequality for $1/2<\gamma<3/2$, its optimal constant and an attaining sech² potential, so it does not cover this paper's matrix-potential Theorem 1.1. [AI-generated per source] [family-level]
Possibly relevant literature (found by the agent, not checked)
- Rupert L. Frank, David Gontier, Mathieu Lewin, The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities (arXiv:2002.04964, 2020). Shows that the scalar best constant is not attained in the one-bound-state case for $\gamma>3/2$ in $d=1$, which bears on why the one-state constant of Theorem 1.1 is claimed only for $1/2<\gamma<3/2$. Not in the bibliography.
- Lukas Schimmer, The state of the Lieb--Thirring conjecture (arXiv:2203.06051, 2022). A survey of known results on optimal Lieb–Thirring constants, including operator-valued versions, against which the matrix claim can be compared. Not in the bibliography.
- Rupert L. Frank, The Lieb–Thirring inequalities: Recent results and open problems (arXiv:2007.09326, 2020-07-18). The survey explains the one-particle and semiclassical constants underlying the paper's claim that the scalar one-state constant remains optimal for every finite matrix size. [family-level]
- Rupert L. Frank, Ari Laptev, Timo Weidl, Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities (Cambridge Studies in Advanced Mathematics 200, 2022). A monograph covering the operator-valued Lieb–Thirring inequalities (Laptev–Weidl, Hundertmark–Laptev–Weidl, Benguria–Loss) that the paper extends to the full open interval. Not in the bibliography.
Digests and expositions
- Latent Seminar, Sharp Lieb–Thirring bounds: one state, matrix channels, and equality (2026-10-07). This exposition distinguishes the three manuscripts and outlines the action inequality, arbitrary-rank trace comparison, and fixed-channel equality argument, explicitly limiting its verification scope. [family-level]
Elsewhere
- Emergent Mind: family 262
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/262.md. The Lean statements have not been checked by people against the paper's claims.
Family 262: Sharp finite-matrix Lieb–Thirring inequalities and all equality cases
Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2\lt \gamma\lt 3/2$ and arbitrary finite-matrix potentials W ≥ 0 with $\int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$: 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.
Other papers in this family:
- Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality · October 5, 2026
- Sharp one-dimensional Lieb–Thirring constants · September 23, 2026