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 Schrödinger operator in \(L^2(\mathbb {R}^n)\) $L^2({\mathbb{R}}^n)$ , these inequalities have applications in quantum mechanics and other fields.