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

What OpenAI claims

Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2 < γ < 3/2 and arbitrary finite-matrix potentials W ≥ 0 with ∫tr (W^γ+1/2) < ∞: 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)

Papers (3)

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and 1/2 < γ < 3/2. For measurable Hermitian positive semidefinite potentials W with ∫_ℝ tr (W^γ+1/2) < ∞, 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.

2. Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials

   5 October 2026 · Preprint, not peer reviewed · PDF

   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 ∫_ℝ tr (W^γ+1/2) < ∞. The matrices may have arbitrary rank and need not commute at different points.

3. Sharp one-dimensional Lieb–Thirring constants

   23 September 2026 · Preprint, not peer reviewed · PDF

   We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every 1/2 < γ < 3/2, the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in L^γ+1/2(ℝ), with all negative eigenvalues included.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.