Sharp finite-matrix Lieb–Thirring inequalities and all equality cases
OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
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Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2< \gamma< 3/2 and arbitrary finite-matrix potentials W ≥ 0 with \int\mathop{tr}\nolimits (W^{\gamma+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 sech^(2) solitons with independent scales and centers, and zero channels.
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Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality
We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and 1/2< \gamma< 3/2. For measurable Hermitian positive semidefinite potentials W with \int_\mathbb R\mathop{tr}\nolimits (W^{\gamma+1/2})< ∞, equality holds precisely for direct sums, in one constant unitary basis,…
Full source abstract
We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and 1/2< \gamma< 3/2. For measurable Hermitian positive semidefinite potentials W with \int_\mathbb R\mathop{tr}\nolimits (W^{\gamma+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.
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Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials
We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent 1/2< \gamma< 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…
Full source abstract
We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent 1/2< \gamma< 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{tr}\nolimits (W^{\gamma+1/2})< ∞. The matrices may have arbitrary rank and need not commute at different points.
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We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every \frac12< \gamma< \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.
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