OpenAI · 262-02 · 2026-10-05 · Mathematical physics

Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials

preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026

Family 262. Sharp one-dimensional Lieb–Thirring inequalities.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

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Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2 < γ < 3/2$ and arbitrary finite-matrix potentials $W \geq 0$ with $\int tr (W^{γ + 1/2}) < \infty$ : 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 $sech^{2}$ solitons with independent scales and centers, and zero channels.

We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent $1/2 < γ < 3/2$ . 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 $\int_{R} tr (W^{γ + 1/2}) < \infty$ . The matrices may have arbitrary rank and need not commute at different points.

Preprint source and Lean scope

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