We present results on the behavior of the dynamic structure function in the short wave length limit using the equation of motion method. Within this framework we study the linear response of a quantum system to an infinitesimal external perturbation by direct minimization of the action integral. As a result we get a set of coupled continuity equations which define the self-energy. We evaluate the self-energy and the dynamic structure function in the short wavelength limit and show that sum rules up to the third moment are fulfilled. This implies, for instance, that the self-energy at short wavelengths and zero frequency is proportional to the kinetic energy per particle. An essential feature in this derivation is that the short range behavior of the two-particle distribution and the long wavelength phonon induced scattering are exactly satisfied. We calculate the condensate fraction and show that our results agree very well with the Monte Carlo simulations.
The momentum distribution ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}$ of atoms in a Bose-condensed system is computed in the long-wavelength limit $p\ensuremath{\rightarrow}0$. We use frequency-moment sum rules for the single-particle Green's function, including a generalized version of Wagner's sum rule appropriate to hard-core potentials. We show that at finite temperatures ($\mathrm{cp}\ensuremath{\ll}{k}_{B}T$) the correction to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}=\frac{{n}_{0}{m}^{2}}{{\ensuremath{\rho}}_{s}{p}^{2}}$ involves the off-diagonal self-energy ${\ensuremath{\Sigma}}_{+\ensuremath{-}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},\ensuremath{\omega}=0)$. In calculating ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}$ in the limit $\mathrm{cp}\ensuremath{\gg}{k}_{B}T$, we include first-sound as well as second-sound contributions.
The dynamic structure function $\mathcal{S}(k, \ensuremath{\omega})$, as a function of wave vector
k and frequency $\ensuremath{\omega}$, is calculated rigorously for a simple Bose gas at
T=0, The dielectric formulation is used to satisfy general symmetry requirements for a Bose system, and the calculation is carried out in the first approximation beyond Bogoliubov taking into account the three-phonon process. An analytic solution in terms of elliptic integrals is obtained for the width of $\mathcal{S}(k, \ensuremath{\omega})$ valid for arbitrary
k and $\ensuremath{\omega}$. Qualitatively, $\mathcal{S}(k, \ensuremath{\omega})$ at finite but small
k is found to have a square-root behavior near the threshold frequency for two-phonon production, a peak at the elementary excitation frequency, and a long power-law tail at high frequencies. Illustrative numerical results are presented for the width of $\mathcal{S}(k, \ensuremath{\omega})$, the imaginary part of the phonon spectrum, and $\mathcal{S}(k, \ensuremath{\omega})$ itself. Finally, the impulse approximation and the extraction of the condensate density
n0 from $\mathcal{S}(k, \ensuremath{\omega})$ in the large-
k limit are discussed.
Through exact numerical solutions we show Bose-Einstein condensation (BEC) for the one-dimensional (1D) bosons with repulsive short-range interactions at zero temperature by taking a particular large size limit. Following the Penrose-Onsager criterion of BEC, we define condensate fraction by the fraction of the largest eigenvalue of the one-particle reduced density matrix. % We show the finite-size scaling behavior such that condensate fraction is given by a scaling function of one-variable: interaction parameter multiplied by a power of particle number. Condensate fraction is nonzero and constant for any large value of particle number or system size, if the interaction parameter is proportional to the negative power of particle number. %Here the interaction parameter is defined by the coupling constant of the delta-function potentials devided by the density. %With the scaling behavior we derive various themodynamic limits where condensate fraction is constant for any large system size; for instance, it is the case even in the system of a finite particle number.
The Bose-Einstein condensation of correlated atoms in a trap is studied by examining the effect of inter-particle correlations to one-body properties of atomic systems at zero temperature using a simplified formula for the correlated two body density distribution. Analytical expressions for the density distribution and rms radius of the atomic systems are derived using four different expressions of Jastrow type correlation function. In one case, in addition, the one-body density matrix, momentum distribution and kinetic energy are calculated analytically, while the natural orbitals and natural occupation numbers are also predicted in this case. Simple approximate expressions for the mean square radius and kinetic energy are also given.
The Bose-Einstein condensation of correlated atoms in a trap is studied by examining the effect of interparticle correlations to one-body properties of atomic systems at zero temperature using a simplified formula for the correlated two-body density distribution. Analytical expressions for the density distribution and rms radius of the atomic systems are derived using four different expressions of the Jastrow-type correlation function. In one case, in addition, the one-body density matrix, momentum distribution and kinetic energy are calculated analytically, while the natural orbitals and natural occupation numbers are also predicted in this case. Simple approximate expressions for the mean-square radius and kinetic energy are also given.