Lieb-Thirring inequalities for the Dirac operator on spheres

TL;DR

This work develops Lieb-Thirring-type bounds for the Dirac operator on spheres $\\mathbb{S}^n$, establishing explicit constants in two inequalities for orthonormal families with zero mean via the positive-spectrum projection $\\Lambda^+(\\slashed D)$. Building on a Clifford-algebra framework and a Weitzenböck-type identity, the authors derive sharp constants $c_n$ and $c'_n$ (with $n=2$ yielding $1/3$) and provide general formulas for all $n\\ge 2$, obtaining corresponding bounds for the classical LT constant $k_{\\mathbb{S}^n}$. The proofs rely on spectral decompositions, localization arguments, and one-dimensional optimization of density functionals $I(\\rho)$, extending techniques from tori and Euclidean settings. The results yield improved upper bounds for the LT constant on $\\mathbb{S}^n$ when $n\\ge 5$ and reveal factorial-type growth, highlighting the interplay between geometry, Dirac spectra, and relativistic spectral bounds on manifolds.

Abstract

In this paper, we obtain bounds for the best constants in two inequalities which can be seen as analogues of the Lieb-Thirring inequality, but with the Dirac operator, on the $n-$sphere. We then apply these results in order to improve the known upper bounds on the classical Lieb-Thirring constant on the $n$-sphere for $n\geq 5$.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 1
• Corollary 3.4
• Theorem 3.5
• Theorem 3.6
• proof : Proof of Theorem \ref{['teos2']}
• proof : Proof of Theorem \ref{['teosn']}