Best constants in Lieb-Thirring inequalities: a numerical investigation

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