Bose-Einstein condensation in a one-dimensional system of interacting bosons

Using the Vakarchuk formulae for the density matrix, we calculate the number N_k of atoms with momentum \hbar k for the ground state of a uniform one-dimensional periodic system of interacting bosons. We obtain for impenetrable point bosons N_0 = 2\sqrt{N} and N_{k=2\pi j/L} = 0.31N_{0}/\sqrt{|j|}. That is, there is no condensate or quasicondensate on low levels at large N. For almost point bosons with weak coupling (\beta=\frac{\nu_{0}m}{\pi^{2}\hbar^{2}n} \ll 1), we obtain N_{0}/N = (\frac{2}{N\sqrt{\beta}})^{\sqrt{\beta}/2} and N_{k=2\pi j/L} = \frac{N_0\sqrt{\beta}}{4|j|^{1-\sqrt{\beta}/2}}. In this case, the quasicondensate exists on the level with k=0 and on low levels with k\neq 0, if N is large and $\beta$ is small (e.g., for N = 10^{10}, \beta = 0.01). A method of measurement of such fragmented quasicondensate is proposed.