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. (Lean)

Manuscripts (3)

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

October 5, 2026 · source · cite as \cite{OAI:Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026}

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.

Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials

October 5, 2026 · source · cite as \cite{OAI:Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026}

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.

Sharp one-dimensional Lieb–Thirring constants

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026}

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

Result 262 · Mathematical physics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.