We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order N −1 (Gross-Pitaevskii regime). We determine the ground state energy and the lowenergy excitation spectrum, up to errors vanishing as N → ∞. Our results confirm Bogoliubov's predictions.

We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in [2], removing the assumption of small interaction potential.

We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. We show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of [37], where fluctuations around the Hartree dynamics for N -particle initial data in the mean-field regime have been analyzed, with ideas from [9], where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.

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 show that the ground state of the system and all states with relatively small excitation energy exhibit complete Bose-Einstein condensation, with a uniform (i.e. N independent) bound on the number of excitations.(1) N,t ϕ 0 ≤ CN α−1 This estimate is non-trivial for all α < 1.Bounds similar to (1.5) have been obtained in [20,12,5,19] for N -boson systems in the mean field limit, described by the Hamilton operator

We consider systems of N bosons in a box with volume one, interacting through a repulsive two-body potential of the form κN 3β−1 V (N β x). For all 0 < β < 1, and for sufficiently small coupling constant κ > 0, we establish the validity of Bogoliubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large N .