Finite-Rank Optimizers for the mass--supercritical Lieb--Thirring and Hardy--Lieb--Thirring Inequalities

We establish the existence of finite-rank operators for an interpolation version of the Lieb–Thirring inequality in the mass–supercritical case, thereby extending a result of Hong, Kwon, and Yoon in 2019 to the full parameter regime. Our method also applies to the Hardy–Lieb–Thirring inequality, where the existence of optimizers faces additional difficulties due to the singularity of the inverse-square potential.