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

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.