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.