The sharp one-dimensional Lieb-Thirring inequality for the sum of eigenvalues

Abstract

We prove that the best constant in the one-dimensional Lieb-Thirring inequality with exponent one is $4/(3\sqrt{3}\pi )$, confirming the Lieb-Thirring conjecture in this case. Moreover, we extend the inequality to operator-valued potentials and obtain the bound $L_{1,d}\le (2/\sqrt{3})L_{1,d}^{\mathrm{cl}}$ in every dimension.

arXiv commentv2: Added Section 2, which gives improved estimates on the Lieb-Thirring constants in higher dimensions