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Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality
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Abstract
We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and $1/2\lt \gamma\lt 3/2$. For measurable Hermitian positive semidefinite potentials W with $\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$, 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.
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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 equality classification. [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 for $\gamma>3/2$ in $d=1$ the best constant is never attained by one bound state, in contrast to the paper's claim that every equality case in $1/2<\gamma<3/2$ has exactly one bound state per nonzero channel. Not in the bibliography.
- Lukas Schimmer, The state of the Lieb--Thirring conjecture (arXiv:2203.06051, 2022). A survey of known optimal constants and optimizers, useful for placing the classification in Theorem 1.1 in context. Not in the bibliography.
- Rupert L. Frank, The Lieb–Thirring inequalities: Recent results and open problems (arXiv:2007.09326, 2020-07-18). The survey's treatment of one-particle optimizers and sharp spectral-moment constants provides scalar background for the paper's classification of matrix extremizers. [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 on the one-dimensional sharp inequalities and their operator-valued versions reviewed in the history section (Hundertmark–Lieb–Thomas, Hundertmark–Laptev–Weidl, Laptev–Weidl, Benguria–Loss). 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: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}
}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:
- Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials · October 5, 2026
- Sharp one-dimensional Lieb–Thirring constants · September 23, 2026