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. G{é}rard, P. Germain and L. Thomann. We then provide another condition so that the second special Hermite function is a global minimizer.