Bogoliubov theory in the Gross-Pitaevskii limit: a simplified approach

Abstract

We show that Bogoliubov theory correctly predicts the low-energy spectral properties of Bose gases in the Gross-Pitaevskii regime. We recover recent results from [6, 7]. While our main strategy is similar to the one developed in [6, 7], we combine it with new ideas, taken in part from [15, 25]; this makes our proof substantially simpler and shorter. As an important step towards the proof of Bogoliubov theory, we show that low-energy states exhibit complete Bose-Einstein condensation with optimal control over the number of orthogonal excitations.

The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime

“…In the context of interacting Bose gases, Bogolubov transformations based on another approximate CCR have been used to study the excitation spectrum; see, for example, [38,23,9,25]. However, for the fermionic problem considered in the present paper, the approximate CCR holds in a very different setting and requires distinct estimation techniques.…”

Large deviations for the ground state of weakly interacting Bose gases

“…In fact [20] proves a central limit theorem for fluctuations around the condensate for the ground state in the Gross-Pitaevski regime. The Gross-Pitaevski scaling regime considers instead of v, the N -dependent two-body interaction potential v β N = N 3β v N (N β •) with β = 1 (for more details and recent progress on results in the Gross-Pitaevski regime see [5,6,8,16]). However, (1.20) follows from adapting the analysis in [20] to the mathematically easier accessible mean-field scaling regime (corresponding to β = 0).…”

The Excitation Spectrum of Two-Dimensional Bose Gases in the Gross–Pitaevskii Regime

“…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 [ 4 , 13 ] 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 [ 8 , 22 ] or by more general trapping potentials [ 15 , 33 ].…”

“…While this strategy is similar to the one used in the three-dimensional setting (see, for example, [ 6 , 8 , 12 , 15 , 22 , 33 ]), the choice of the appropriate unitary transformations and their action strongly depend on the specific problem under consideration.…”

“…In particular, since cubic terms in the Hamilton operator carry large contributions to the energy (growing with N , as ) we are not able to prove a-priori bounds on moments of the number of excitations (nor on products of the energy with moments of the number of excitations operator), which were important in the three dimensional setting [ 8 ]. To overcome this problem, we are going to apply a localization on the number of particle argument (similarly to the one recently exploited in [ 22 , 33 ]), combined with a-priori bounds on the energy of the excitations. A second important difference, compared with the three-dimensional setting, is that even after quadratic and cubic conjugations, the quartic part of the (renormalized) excitation Hamiltonian is not negligible on uncorrelated states.…”