Lieb-Thirring inequalities with improved constants

TL;DR

This paper improves the constants in Lieb-Thirring inequalities for multi-dimensional Schrödinger operators by deriving a matrix version of a Sobolev inequality, enhancing the bounds on sums of negative eigenvalues.

Contribution

It introduces a matrix-based Sobolev inequality to refine the known constants in Lieb-Thirring inequalities for multi-dimensional cases.

Findings

Improved constants in Lieb-Thirring inequalities.

Enhanced bounds for sums of negative eigenvalues.

Extension of Sobolev inequalities to matrix form.

Abstract

Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schroedinger operators.