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

Giao Ky Duong, Thi Minh Thao Le, Phan Thành Nam, Phuoc-Tai Nguyen

TL;DR

This paper proves the existence of finite-rank optimizers for the mass–supercritical Lieb–Thirring interpolation and the Hardy–Lieb–Thirring interpolation, extending HKY2019 to the full parameter range. The authors first solve the constrained (finite-trace) problems and then derive uniform estimates to remove the constraint, yielding global finite-rank optimizers represented as finite sums of rank-one projections and governed by Euler–Lagrange equations. The Hardy potential introduces additional challenges, which are overcome by CLR-type bounds for the Hardy operator and a careful radial/non-radial analysis. The results are established in the fractional Laplacian setting and rely on concentration-compactness, IMS formulas, and ground-state representations to secure compactness and structure of minimizers. These findings advance the understanding of optimizer existence for density-based interpolation inequalities and have potential implications for density-functional-type theories in quantum systems.

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.

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 the mass–supercritical Lieb–Thirring interpolation and the Hardy–Lieb–Thirring interpolation, extending HKY2019 to the full parameter range. The authors first solve the constrained (finite-trace) problems and then derive uniform estimates to remove the constraint, yielding global finite-rank optimizers represented as finite sums of rank-one projections and governed by Euler–Lagrange equations. The Hardy potential introduces additional challenges, which are overcome by CLR-type bounds for the Hardy operator and a careful radial/non-radial analysis. The results are established in the fractional Laplacian setting and rely on concentration-compactness, IMS formulas, and ground-state representations to secure compactness and structure of minimizers. These findings advance the understanding of optimizer existence for density-based interpolation inequalities and have potential implications for density-functional-type theories in quantum systems.

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.

Paper Structure

This paper contains 13 sections, 16 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Mass--supercritical Lieb--Thirring inequality
  4. Mass--supercritical Hardy--Lieb--Thirring inequality
  5. Proof of Theorem \ref{['thm:optimizer-LT']}
  6. Existence of optimizers for the restricted problem
  7. Structure of the minimizer
  8. Conclusion of Theorem \ref{['thm:optimizer-LT']}
  9. Proof of Theorem \ref{['thm:optfinite']}
  10. Hardy--Lieb--Thirring inequality
  11. Optimizers for the restricted problem
  12. Concentration compactness lemmas
  13. IMS formulas

Key Result

Theorem 2.1

Let $d \ge 1$, $0 < s \le 1$, $s<d/2$ and $q \in \left(\tfrac{d+2s}{d}, \tfrac{d}{d-2s}\right)$. Then the Lieb--Thirring interpolation inequality LT-inequality has a finite-rank optimizer $0 \le \gamma_\infty \le 1$. This optimizer can be represented in the form $\gamma_\infty = \sum_{n=1}^M |u_n\r with $-\mu_1 \leq -\mu_2 < \cdots \le - \mu_M \le 0$ denoting the lowest eigenvalues (counting mult

Theorems & Definitions (32)

  • Conjecture 1.1: Optimizers of Lieb--Thirring interpolation inequalities
  • Theorem 2.1: Finite-rank optimizers for the mass--supercritical Lieb--Thirring inequality
  • Theorem 2.2: Finite-rank optimizers for the mass--supercritical Hardy--Lieb--Thirring inequality
  • Lemma 3.1: Existence of optimizers
  • proof
  • Lemma 3.2: Spectral decomposition
  • Remark 3.3
  • Lemma 3.4
  • proof : Proof of Lemma \ref{['lmm:mimimizerI-hardy']}
  • Lemma 3.5
  • ...and 22 more