---
_id: '2356'
abstract:
- lang: eng
text: 'Recent experimental and theoretical work has shown that there are conditions
in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function
Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical
phenomenon because it is not necessary to have a trap width that is the size of
an atom - as might have been supposed - but it suffices merely to have a trap
width such that the energy gap for motion in the transverse direction is large
compared to the energy associated with the motion along the trap. Up to now the
theoretical arguments have been based on variational - perturbative ideas or numerical
investigations. In contrast, this paper gives a rigorous proof of the one-dimensional
behavior as far as the ground state energy and particle density are concerned.
There are four parameters involved: the particle number, N, transverse and longitudinal
dimensions of the trap, r and L, and the scattering length a of the interaction
potential. Our main result is that if r/L → 0 and N → ∞ the ground state energy
and density can be obtained by minimizing a one-dimensional density functional
involving the Lieb-Liniger energy density with coupling constant ∼ a/r 2. This
density functional simplifies in various limiting cases and we identify five asymptotic
parameter regions altogether. Three of these, corresponding to the weak coupling
regime, can also be obtained as limits of a three-dimensional Gross-Pitaevskii
theory. We also show that Bose-Einstein condensation in the ground state persists
in a part of this regime. In the strong coupling regime the longitudinal motion
of the particles is strongly correlated. The Gross-Pitaevskii description is not
valid in this regime and new mathematical methods come into play.'
author:
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jakob
full_name: Yngvason, Jakob
last_name: Yngvason
citation:
ama: Lieb É, Seiringer R, Yngvason J. One-dimensional behavior of dilute, trapped
Bose gases. *Communications in Mathematical Physics*. 2004;244(2):347-393.
doi:10.1007/s00220-003-0993-3
apa: Lieb, É., Seiringer, R., & Yngvason, J. (2004). One-dimensional behavior
of dilute, trapped Bose gases. *Communications in Mathematical Physics*.
Springer. https://doi.org/10.1007/s00220-003-0993-3
chicago: Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “One-Dimensional Behavior
of Dilute, Trapped Bose Gases.” *Communications in Mathematical Physics*.
Springer, 2004. https://doi.org/10.1007/s00220-003-0993-3.
ieee: É. Lieb, R. Seiringer, and J. Yngvason, “One-dimensional behavior of dilute,
trapped Bose gases,” *Communications in Mathematical Physics*, vol. 244,
no. 2. Springer, pp. 347–393, 2004.
ista: Lieb É, Seiringer R, Yngvason J. 2004. One-dimensional behavior of dilute,
trapped Bose gases. Communications in Mathematical Physics. 244(2), 347–393.
mla: Lieb, Élliott, et al. “One-Dimensional Behavior of Dilute, Trapped Bose Gases.”
*Communications in Mathematical Physics*, vol. 244, no. 2, Springer, 2004,
pp. 347–93, doi:10.1007/s00220-003-0993-3.
short: É. Lieb, R. Seiringer, J. Yngvason, Communications in Mathematical Physics
244 (2004) 347–393.
date_created: 2018-12-11T11:57:11Z
date_published: 2004-01-01T00:00:00Z
date_updated: 2021-01-12T06:56:59Z
day: '01'
doi: 10.1007/s00220-003-0993-3
extern: 1
intvolume: ' 244'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0305025
month: '01'
oa: 1
page: 347 - 393
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4569'
quality_controlled: 0
status: public
title: One-dimensional behavior of dilute, trapped Bose gases
type: journal_article
volume: 244
year: '2004'
...