Complete Bose–Einstein Condensation in the Gross–Pitaevskii Regime

We consider a gas of N bosons in a box with volume one interacting through a two-body potential with scattering length of order N- 1 (Gross–Pitaevskii limit). Assuming the (unscaled) potential to be sufficiently weak, we prove complete Bose–Einstein condensation for the ground state and for many-body states with finite excitation energy in the limit of large N with a uniform (N-independent) bound on the number of excitations.