# A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas

@article{Lieb2000ARD, title={A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas}, author={Elliott H. Lieb and Robert Seiringer and Jakob Yngvason}, journal={Communications in Mathematical Physics}, year={2000}, volume={224}, pages={17-31} }

Abstract: We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number N is large but ρ̅a2 is small, where ρ̅ is the average particle density and a the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross–Pitaevskii (GP) energy functional with a coupling constant g~1/|1n(ρ̅a2)|. In…

## 154 Citations

One-Dimensional Behavior of Dilute, Trapped Bose Gases

- Physics
- 2004

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…

The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases

- Physics, Mathematics
- 2014

We study the ground state of a trapped Bose gas, starting from the full many-body Schrodinger Hamiltonian, and derive the nonlinear Schrodinger energy functional in the limit of large particle…

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

- Physics
- 2012

We study the ground state which minimizes a Gross–Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the Thomas–Fermi limit where a small parameter…

Ground state asymptotics of a dilute, rotating gas

- Physics
- 2003

We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the…

Self-consistent calculation of the coupling constant in the Gross-Pitaevskii equation

- Physics
- 2004

A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling…

Rapidly rotating Bose-Einstein condensates in strongly anharmonic traps

- Physics
- 2007

We study a rotating Bose-Einstein condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of two-dimensional Gross-Pitaevskii theory. We write the coupling constant…

Gross-Pitaevskii Theory of the Rotating Bose Gas

- Physics
- 2002

Abstract: We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely,…

Ground-State Energy of a Dilute Fermi Gas

- Physics
- 2005

Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about…

Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground State Energy Formula

- Physics
- 2001

Recent experimental breakthroughs in the treatment of dilute Bose gases have renewed interest in their quantum mechanical description, respectively in approximations to it. The ground state…

ON THE ENERGY OF A BOSE–EINSTEIN CONDENSATE IN AN OPTICAL LATTICE

- Mathematics, Physics
- 2007

In this paper, we study the Gross–Pitaevskii energy of a Bose–Einstein condensate in the presence of an optical lattice, modeled by a periodic potential V(x3) in the third direction. We study a…

## References

SHOWING 1-10 OF 25 REFERENCES

The Ground State Energy of a Dilute Two-Dimensional Bose Gas

- Physics
- 2000

The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be E0/N=(2πℏ2ρ/m)|ln(ρa2)|−1, to leading order, with a…

Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional

- Physics
- 2000

The ground-state properties of interacting Host gases in external potentials, as considered in recent exptriments, are usually described by means of the Gross-Pitaevskii energy functional. We present…

Bose-Einstein condensation in a two-dimensional, trapped, interacting gas

- Physics
- 1998

The observation of the Bose-Einstein condensation ~BEC! phenomenon in dilute atomic gases @1‐4# has caused a lot of attention, because it provides opportunities to study the thermodynamics of weakly…

Renormalization-group analysis of the ground-state properties of dilute Bose systems in d spatial dimensions.

- PhysicsPhysical review. B, Condensed matter
- 1992

A low-density system of Bose particles of mass m and density n interacting through a short-range potential with range a is considered in d dimensions using a renormalization approach. The expansion…

Quantum-Monte-Carlo Calculations for Bosons in a Two-Dimensional Harmonic Trap

- Physics
- 1998

Path-Integral-Monte-Carlo simulation has been used to calculate the properties of a two-dimensional (2D) interacting Bose system. The bosons interact with hard-core potentials and are confined to a…

Dilute Bose gas in two dimensions.

- PhysicsPhysical review. B, Condensed matter
- 1988

An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group.

Low-dimensional bose liquids: beyond the gross-pitaevskii approximation

- Physics
- 2000

The Gross-Pitaevskii approximation is a long-wavelength theory widely used to describe a variety of properties of dilute Bose condensates, in particular trapped alkali gases. We point out that for…

Ground State Energy of the Low Density Bose Gas

- Physics
- 1997

Now that the properties of low temperature Bose gases at low density, $\rho$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago.…

Two-Dimensional Gross-Pitaevskii Equation: Theory of Bose-Einstein Condensation and the Vortex State

- Physics
- 1999

We derive the Gross-Pitaevskii equation in two-dimension from the first principles of two-dimensional scattering theory. Numerical calculation of the condensate wave function shows that atoms in a…

Bose-Einstein condensation in a two-dimensional trap

- Physics
- 2000

The theory of Bose-Einstein condensation in a two-dimensional(2D) harmonic trap is developed from 2D Gross-Pitaevskii equation. The 2D interaction strength is obtained from a 2D collision theory.
We…