ua nt - p h / 05 09 07 2 v 2 6 D ec 2 00 6 Note on the Zero - Energy - Limit Solution for the Modified Gross - Pitaevskii Equation

The modified Gross-Pitaevskii equation was derived and solved to obtain the 1D solution in the zero-energy limit. This stationary solution could account for the dominated contributions due to the kinetic effect as well as the chemical potential in inhomogeneous Bose gases. Studies of collision phenomena in rather cold gases, e.g., dilute Bose gases, have recently attracted many researchers' attention [1-2]. One relevant research interest is about the solution of the appropriate and modified Gross-Pitaevskii equations for different dimensions [3-4]. New possibilities for observation of macroscopic quantum phenomena arises because of the recent realization of Bose-Einstein condensation in atomic gases [1-2]. There are two important features of the system-weak interaction and significant spatial inhomogeneity. Because of this inhomo-geneity a non-trivial zeroth-order theory exists, compared to the first-order Bogoliubov theory. This theory is based on the mean-field Gross-Pitaevskii equation for the condensate ψ-function. The equation is classical in its essence but contains the (h/2p) constant explicitly. Phenomena such as collective modes, interference, tunneling, Josephson-like current and quantized vortex lines can be described using this equation. The study of deviations from the zeroth-order theory arising from zero-point and thermal fluctuations is also of great interest [5-7]. Thermal fluctuations are described by elementary excitations which define the thermodynamic behaviour of the system and result in Landau-type damping of collective modes. As a preliminary attempt, followling the mean-field approximate formulation in [3], in this letter , we plan to investigate the 1D solution for the modified Gross-Pitaevski equation in the zero-energy limit. This presentation will give more clues to the studies of the quantum non-equilibrium thermodynamics in inhomogeneous (dilute) Bose gases and the possible appearance of the kinetic mechanism before and/or after Bose-Einstein condensation which is directly linked to the particles (number) density and their energy states or chemical potentials. The generalization of the Bogoliubov prescription [8] for the ψ-operator to the case of a spatially nonuniform system is ˆ ψ(r, t) ≈ ψ 0 (r, t) + ˆ φ(r, t) (1)