Series Expansion in the Variational Approach to Many-Boson Problems

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 $g\left(\mathcal{r}\right)$ differs little from its asymptotic value. The treatment of the problem is based on the development of power series in $\alpha =1-g\left(0\right)\mathrm{}$ for all physical quantities which depend on the particle density. The $n$-particle distribution functions $p^{\mathrm{}\left(n\right)\mathrm{}}$ are evaluated to order ${\alpha }^4$ as functionals in the $g\left(\mathcal{r}\right)$ function for $n=3\mathrm{} \mathrm{and} 4$ 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 $p^{\mathrm{}\left(3\right)\mathrm{}}$ function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\alpha }^4$, for the two-particle correlation function $\mathcal{U}\left(\mathcal{r}\right)$, and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for $\mathcal{U}\left(\mathcal{r}\right)$ yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of $\mathcal{U}\left(\mathcal{r}\right)$, $p^{\mathrm{}\left(3\right)\mathrm{}}$, $p^{\mathrm{}\left(4\right)\mathrm{}}$, and the ground-state energy are computed using the Gaussian approximation for $g\left(\mathcal{r}\right)$, 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 $p^{\mathrm{}\left(n\right)\mathrm{}}$ in terms of $g\left(\mathcal{r}\right)$.