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 semidefinite potential W satisfying ${\int }_{\mathbb{R}}\mathrm{tr}(W^{\gamma +1/2})<\infty$. The matrices may have arbitrary rank and need not commute at different points.