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.
The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1$ and the latter by $\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1$, where
g(r) is the radial distribution function and
N0 is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\ensuremath{\alpha}$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\ensuremath{\beta}\ensuremath{\ll}1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\ensuremath{\delta}$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.
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 an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.
We study many-body correlations in the ground state of a general quantum system of bosons or fermions by including an additional Jastrow function in our recently proposed variational coupled-cluster method. Our approach combines the advantages of state-dependent correlations in the coupled-cluster theory and of the strong, short-ranged correlations of the Jastrow function. We apply a generalized linked-cluster expansion for the Jastrow wave function and provide a detailed analysis for practical evaluation of the Hamiltonian expectation value as an energy functional of the Jastrow function and the bare density-distribution functions introduced and calculated in our earlier publications; a simple, first-order energy functional is derived and detailed formulas for the higher-order contributions are provided. Our energy functional does not suffer the divergence as most coupled-cluster calculations often do when applying to Hamiltonians with hardcore potentials. We also discuss possible applications of our technique, including applications to strongly correlated fermion systems.