Acta Mathematica Scientia

MEAN-FIELD LIMIT OF BOSE-EINSTEIN CONDENSATES WITH ATTRACTIVE INTERACTIONS IN R^2

郭玉劲, 陆璐

Published Jan 1, 2016• 0 citations

Abstract

Starting with the many-body Schrdinger Hamiltonian in R~2, we prove that the ground state energy of a two-dimensional interacting Bose gas with the pairwise attractive interaction approaches to the minimum of the Gross-Pitaevskii energy functional in the meanfield regime, as the particle number N→∞ and however the scattering length κ→0. By fixing N|κ|, this leads to the mean-field approximation of Bose-Einstein condensates with attractive interactions in R~2.

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Acta Mathematica Scientia

MEAN-FIELD LIMIT OF BOSE-EINSTEIN CONDENSATES WITH ATTRACTIVE INTERACTIONS IN R^2

郭玉劲, 陆璐

Published Jan 1, 2016• 0 citations

OpenAlex

Abstract

Starting with the many-body Schrdinger Hamiltonian in R2, we prove that the ground state energy of a two-dimensional interacting Bose gas with the pairwise attractive interaction approaches to the minimum of the Gross-Pitaevskii energy functional in the meanfield regime, as the particle number N→∞ and however the scattering length κ→0. By fixing N|κ|, this leads to the mean-field approximation of Bose-Einstein condensates with attractive interactions in R2.

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Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order

N^{-1+\kappa }

N - 1 + κ , for

\kappa >0

κ > 0 . Assuming that

\kappa \in (0;1/43)

κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

We consider many-body effects on particle scattering in one-, two-, and three-dimensional (3D) Bose gases. We show that at

T=0

these effects can be modeled by the simpler two-body T matrix evaluated off the energy shell. This is important in 1D and 2D because the two-body T matrix vanishes at zero energy and so mean-field effects on particle energies must be taken into account to obtain a self-consistent treatment of low-energy collisions. Using the off-shell two-body T matrix we obtain the energy and density dependence of the effective interaction in 1D and 2D and the appropriate Gross-Pitaevskii equations for these dimensions. Our results provide an alternative derivation of those of Kolomeisky and co-workers. We present numerical solutions of the Gross-Pitaevskii equation for a 2D condensate of hard-sphere bosons in a trap. We find that the interaction strength is much greater in 2D than for a 3D gas with the same hard-sphere radius. The Thomas-Fermi regime is, therefore, approached at lower condensate populations and the energy required to create vortices is lowered compared to the 3D case.