Bogoliubov Theory for Trapped Bosons in the Gross–Pitaevskii Regime

Bogoliubov Theory in the Gross-Pitaevskii Limit: a Simplified Approach

“…Recently, a rigorous version of Bogoliubov theory [8] has been developed in [4,5,6,7] to provide more precise information on the low-energy spectrum of (1), resolving the ground state energy and the low-lying excitations up to errors that vanish in the limit N Ñ 8, and on the corresponding eigenvectors, showing Bose-Einstein condensation with optimal control on the number of orthogonal excitations. Analogous results have been established also for Bose gases trapped by external potentials in the Gross-Pitaevskii regime [23,10,25,11] and for Bose gases in scaling limits interpolating between the Gross-Pitaevskii regime and the thermodynamic limit [1,9]. Very recently, the upper bound for the ground state energy has been also extended to the case of hard-sphere interaction, as announced in [2].…”

“…hence we only have to bound }|p| ´2p VN ˚ϕq} 1 . Iterating (11) and using the regularizing estimate }|p| ´2 VN ˚g} 6p{p6`pq`ε ď C ε } VN } 2 }g} p for all ε ą 0, p ě 6{5, g P ℓ p pΛ ˚q and some C ε ą 0, we obtain that }ϕ} 1 ă 8. Separating high and low momenta, we obtain for A ě 1 and ε ą 0,…”

A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime

“…For V ∈ L 3 (R 3 ), more precise information on the low-energy spectrum of (1.1) have been determined in [4]. Here, the ground state energy was proven to satisfy Recently, (1.3), (1.5) have been also extended to the non-homogeneous case of Bose gases trapped by external fields in [24,7]. While the approach of [24] applies to V ∈ L 1 (R 3 ), the validity of (1.3), (1.5) for bosons interacting through non-integrable potentials is still an open question.…”

The excitation spectrum of two dimensional Bose gases in the Gross-Pitaevskii regime

“…i) To keep our analysis as simple as possible, we restrict out attention to bosons moving in the two-dimensional unit torus. Our results could be extended to more general trapping potentials, combining the proof of Theorem 1.1 with ideas from [30,13,31,14], recently developed in the three dimensional setting.…”

“…The proof of Theorem 1.1 is based on Fock space methods, recently developed in the three-dimensional setting, to study the dynamics of Bose-Einstein condensates [3,12] and to investigate the equilibrium properties of dilute gases in the Gross-Pitaevskii regime. In particular, these techniques led to the verification of the predictions of Bogoliubov theory for the ground state energy and the excitation spectrum of three dimensional Bose gas in the Gross-Pitaevskii regime, confined on the unit torus [7,20] or by more general trapping potentials [14,31].…”