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

TL;DR

This paper provides a direct proof for the existence of ground state minimizers of the Bogoliubov energy functional at zero temperature, applicable to a broader class of potentials, and shows that over half of the particles are in the condensate.

Contribution

It offers a new, direct proof of minimizer existence for the Bogoliubov functional, extending applicability and revealing condensate properties.

Findings

Existence of minimizers at zero temperature proven directly.

Broader class of interaction potentials covered.

More than half of particles are in the Bose-Einstein condensate in ground states.

Abstract

The Bogoliubov energy functional proposed recently by Napi\'orkowski, 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.