The ground state of a one-dimensional system of many bosons interacting through a repulsive $\ensuremath{\delta}$-function potential is studied when the coupling parameter $\ensuremath{\gamma}$ of the system is small. The procedure is based on the variational analysis in the Bijl-Dingle-Jastrow wave-function space. The ground-state energy, determined in a series form in powers of ${\ensuremath{\gamma}}^{\frac{1}{2}}$ and truncated after the third leading term, is found to be in fair agreement with exact results of Lieb and Liniger over a range of the coupling parameter extending up to $\ensuremath{\gamma}=6$. Two slightly different approaches are considered, and they are shown to be equivalent.
The ground-state energy of a one-dimensional system of many bosons interacting via a repulsive s-function potential is carried out through third-order corrections, when the coupling parameter γ of the system is small. The procedures' are basically developed following both the Bogoliubov-Zubarev and the Davison-Feenberg methods. It is seen numerically that a slight difference exists in between, at least, starting from the third-order correction; the former one gives rise to a more improved agreement with the exact results obtained by Lieb and Liniger, and also, the variational contribution in the later is still dominant even in the third-order correction.
The method of self-consistent-field approximation is applied for the study of the ground-state properties of a one-dimensional system of bosons, of density $\ensuremath{\rho}$, first interacting through finite rigid rods, of length
l, and again via a repulsive $\ensuremath{\delta}$-function potential. We obtain the energy spectrum and the sound velocity. The results are close to the exact one for $\ensuremath{\rho}l<0.5$. The energy spectrum obtained shows a rotonlike region which is qualitatively similar to the two- and three-dimensional versions. For a Bose gas interacting through a repulsive $\ensuremath{\epsilon}$ potential of strength
2c, we derive the energy spectrum, the sound velocity, and the ground-state energy. The results are compared with Girardeau and Lieb's exact expressions. In the small-coupling limit our calculations show the same results as obtained by Bogoliubov's perturbation theory. The ground-state energy obtained for small $\ensuremath{\gamma}=\frac{c}{\ensuremath{\rho}}$ is found to agree closely with several different works found in the literature. Unfortunately our theory diverges for the ground-state energy in the limit $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The sound velocity in this limit is finite but twice as small as from Girardeau's exact results.
The ground state of a many-boson system is studied within the range of the Bijl-Dingle-Jastrow-type description when the radial distribution function
g(r) differs little from its asymptotic value. The treatment of the problem is based on the development of power series in $\ensuremath{\alpha}=1\ensuremath{-}g(0)$ for all physical quantities which depend on the particle density. The
n-particle distribution functions
p(n) are evaluated to order ${\ensuremath{\alpha}}^{4}$ as functionals in the
g(r) function for
n=3and4 using the cluster-expansion procedure outlined by Abe. These results are used in connection with the improvement of the ground-state description when the wave function is not the optimum choice. Using
p(3) function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\ensuremath{\alpha}}^{4}$, for the two-particle correlation function
U(r), and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for
U(r) yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of
U(r),
p(3),
p(4), and the ground-state energy are computed using the Gaussian approximation for
g(r), and the results show that the errors associated with the HNC approximation are small. A brief discussion is presented on the method of determining the general expansion coefficients of the correlation functions of
p(n) in terms of
g(r).
An occupation-number formulation is considered for a many-particle boson system when the ground state is described by the Bijl-Dingle-Jastrow (BDJ) type of variational wave function. A nonunitary operator acting on the state function of a noninteracting boson system is introduced to convert the BDJ function in configuration space into an equivalent state function in second-quantized space. The momentum distribution function is derived in an infinite series in terms of the liquid-structure function and the correlation function. In the case of the optimum variational description under the uniform limit, the leading term of the obtained result is found to agree with the Bogoliubov formula evaluated in the weak-coupling limit. The same analysis in the occupation-number space also reproduces the mean kinetic energy calculated in configuration space.