A Simple 2nd Order Lower Bound to the Energy of Dilute Bose Gases

Abstract

For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound e(ρ) ≥ 4πaρ 2 (1 − C ρa 3 ). e( ρ ) ≥ 4π ρ 2 a 1 − C ρa 3 .( 1.4)Our result is the first rigorous lower bound on the hard core potential that gives the correct order for the correction term. (See below for a further discussion of the expected correction term, the so-called Lee-Huang-Yang term [9]).

The Second-Order Correction to the Ground State Energy of the Dilute Bose Gas

“…In the accompanying paper [23] we give a second order lower bound on the ground state energy for the unscaled setting (R = a), which is consistent with (6) but does not capture the sharp constant.…”

“…In [23] we modify the localization of the kinetic energy, thereby avoiding an error term originating from (A.15), which would not be compatible with the LHY-order in the unscaled setting of [23]. We expect that a corresponding modification in the present paper would allow us to state Assumption 2 with η = 1 30 instead of η < 1 30 .…”

Bogoliubov theory for dilute Bose gases: The Gross-Pitaevskii regime

“…• It is well known that the ground state energy of bosons in the Gross-Pitaevskii regime are characterized by a correlation structure which varies on the length scale of the scattering length of the interaction a N ∼ N −1 , and which can be modeled by solution of the zero energy scattering equation. This is the key ingredient to show upper and lower bounds consistent with (1.11) at leading order [35], and to establish the results in [16,20,50]. The same correlation structure has to be included in any approach aimed to show the emergence of the Gross-Pitaevskii equation as an effective description for the evolution of initially trapped Bose-Einstein condensates which evolves under the dynamics generated by (1.9) (see [6] and references therein).…”

“…An upper bound for the ground state energy in the thermodynamic limit coinciding with (1.5) up to second order was established in [50] (improving a previous result by [20]). Very recently, a lower bound for the ground state energy which establish the correct order of the next to leading order contribution in the whole dilute regime, has been obtained by [16], but without control on the constant.…”

The excitation spectrum of the Bose gas in the Gross–Pitaevskii regime

“…Bogoliubov's ideas have been implemented for the derivation of the ground state energy of the Bose gas in the thermodynamic limit (the Lee-Huang-Yang formula) [14,8,9,13], and before for the computation of the ground state energy of the bosonic jellium [21]. For the Bose gas in the mean field regime more information is available, and Bogoliubov's method has been implemented to give the excitation spectrum (see [27,15,17,10,26]).…”