Lieb-Thirring inequalities on the spheres and $SO(3)$

TL;DR

This paper derives improved upper bounds for Lieb-Thirring inequalities on spheres of dimension greater than 2 and on the group SO(3), with potential applications in quantum mechanics.

Contribution

It provides new, explicit upper bounds for Lieb-Thirring inequalities on spheres and SO(3), extending to general compact Lie groups.

Findings

Improved upper bounds for spheres of dimension > 2

Explicit upper bound for Lieb-Thirring on SO(3)

Discussion of estimates for general compact Lie groups

Abstract

In this paper, we obtain new upper bounds for the Lieb-Thirring inequality on the spheres of any dimension greater than $2$. As far as we have checked, our results improve previous results found in the literature for all dimensions greater than $2$. We also prove and exhibit an explicit new upper bound for the Lieb-Thirring inequality on $SO(3)$. We also discuss these estimates in the case of general compact Lie groups. Originally developed for estimating the sums of moments of negative eigenvalues of the Schr\"odinger operator in $L^2(\mathbb{R}^n)$, these inequalities have applications in quantum mechanics and other fields.