Lieb-Thirring inequalities for the Dirac operator on spheres

TL;DR

This paper establishes bounds for Lieb-Thirring type inequalities involving the Dirac operator on spheres and uses these to improve known constants for the classical Lieb-Thirring inequality in higher dimensions.

Contribution

It introduces new bounds for the best constants in Dirac operator inequalities on spheres and enhances existing bounds for the Lieb-Thirring constant in dimensions five and above.

Findings

Derived bounds for Dirac operator inequalities on spheres

Improved upper bounds for Lieb-Thirring constants in dimensions n≥5

Extended the applicability of Lieb-Thirring inequalities to spherical geometries

Abstract

In this paper, we obtain bounds for the best constants in two inequalities which can be seen as analogues of the Lieb-Thirring inequality, but with the Dirac operator, on the $n-$sphere. We then apply these results in order to improve the known upper bounds on the classical Lieb-Thirring constant on the $n$-sphere for $n\geq 5$.