Best constants in Lieb-Thirring inequalities: a numerical investigation

TL;DR

This paper numerically investigates the optimal constants in Lieb-Thirring inequalities, providing new lower bounds, insights into maximizers, and formulating a comprehensive conjecture for all parameter values.

Contribution

It introduces a numerical approach using a fixed-point algorithm and finite element discretization to estimate best constants and formulate a complete conjecture.

Findings

New lower bounds for Lieb-Thirring constants

Insights into the behavior of maximizers

Confirmation of some existing conjectures

Abstract

We investigate numerically the optimal constants in Lieb-Thirring inequalities by studying the associated maximization problem. We use a monotonic fixed-point algorithm and a finite element discretization to obtain trial potentials which provide lower bounds on the optimal constants. We examine the one-dimensional and radial cases in detail. Our numerical results provide new lower bounds, insight into the behavior of the maximizers and confirm some existing conjectures. Based on our numerical results, we formulate a complete conjecture about the best constants for all possible values of the parameters.