Sharp Lieb-Thirring Inequalities in High Dimensions

TL;DR

This paper derives sharp Lieb-Thirring inequalities in high dimensions using a matrix trace formula approach, extending known results with optimal constants for certain parameter ranges.

Contribution

It introduces a matrix version of the trace formula that yields sharp Lieb-Thirring inequalities for higher dimensions and specific exponents, improving previous bounds.

Findings

Established sharp Lieb-Thirring inequalities with optimal constants

Extended trace formula methods to matrix operators in high dimensions

Applicable for exponents /2 e2 a0 e2 a4 a0 e2 a4 a0 e2 a4 a0 e2 a4 a0 e2 a4 a0 e2 a4 a0 in arbitrary dimensions

Abstract

We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{\gamma,d}$ with $\gamma\ge 3/2$ and arbitrary $d\ge 1$. (revised, to appear in Acta Math)