# Lieb-Thirring inequality with semiclassical constant and gradient error term

## TL;DR

This paper improves the Lieb-Thirring inequality by establishing a lower bound on kinetic energy that matches the semiclassical constant and includes a lower-order gradient error term, advancing understanding of fermionic energy bounds.

## Contribution

The authors prove an improved Lieb-Thirring inequality with the semiclassical constant and a lower-order gradient error term, addressing a long-standing open problem.

## Key findings

- Established a lower bound matching the semiclassical constant
- Introduced a gradient error term of lower order
- Enhanced the understanding of kinetic energy bounds for fermions

## Abstract

In 1975, Lieb and Thirring derived a semiclassical lower bound on the kinetic energy for fermions, which agrees with the Thomas-Fermi approximation up to a constant factor. Whenever the optimal constant in their bound coincides with the semiclassical one is a long-standing open question. We prove an improved bound with the semiclassical constant and a gradient error term which is of lower order.