An Optimal Minimization Problem in the Lowest Landau Level and Related Questions

We solve a minimization problem related to the cubic Lowest Landau level equation, which is used in the study of Bose–Einstein condensation. We provide an optimal condition for the Gaussian to be the unique global minimizer. This extends previous results from P. Gérard, P. Germain and L. Thomann. We then provide another condition so that the second special Hermite function is a global minimizer.