Density matrices of a many-Boson system at low temperatures

By means of integration over collective variables in the Penrose formula for the $N$-particle density matrix, the explicit expressions for the s-particle matrices $(s/N\to 0)$ are obtained. The density matrices possess an exponential form and this ensures physically correct behaviour of distribution functions at short distances as well as large ones. The first two terms of the expansion reproduce the result of the Bogoliubov theory. The particle momentum distribution, the pair distribution function and the average energy are investigated. The numerical results for the known models of the Bose-gas are obtained. For the one-dimensional Bose-gas the simple expression for the ground state energy as function of the coupling parameter $\gamma$ is obtained. This formula is exact in the weak coupling limit, $\gamma\to 0$ and gives qualitatively correct result for the behaviour in the limit $\gamma\to\infty$.