On Ground States of the Bogoliubov Energy Functional: A Direct Proof

The Bogoliubov energy functional proposed recently by Napiórkowski, Reuvers and Solovej is revisited. We offer a direct proof of the existence of minimizers at zero temperature, which covers a significantly larger class of interaction potentials. The ideas used in this proof also imply that in any ground state, more than half of the particles are inside the Bose-Einstein condensate.