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

TL;DR

This paper proves the existence of finite-rank optimizers for a generalized Lieb--Thirring inequality in the mass--supercritical case, extending previous results and addressing challenges posed by singular potentials.

Contribution

It extends the existence of finite-rank optimizers to the full parameter regime for the interpolation Lieb--Thirring inequality and applies the method to the Hardy--Lieb--Thirring inequality.

Findings

Existence of finite-rank operators in the mass--supercritical regime

Extension of previous results to full parameter range

Application to Hardy--Lieb--Thirring inequality with singular potentials

Abstract

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.