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The exact ground state of the many-body Schr\"odinger equation for $N$ bosonson a one-dimensional ring interacting via pairwise $\delta$-functioninteraction is presented for up to fifty particles. The solutions are obtainedby solving Lieb and Liniger's system of coupled transcendental equations forfinite $N$. The ground state energies for repulsive and attractive interactionare shown to be smoothly connected at the point of zero interaction strength,implying that the \emph{Bethe-ansatz} can be used also for attractiveinteraction for all cases studied. For repulsive interaction the exact energiesare compared to (i) Lieb and Liniger's thermodynamic limit solution and (ii)the Tonks-Girardeau gas limit. It is found that the energy of the thermodynamiclimit solution can differ substantially from that of the exact solution forfinite $N$ when the interaction is weak or when $N$ is small. A simple relationbetween the Tonks-Girardeau gas limit and the solution for finite interactionstrength is revealed. For attractive interaction we find that the true groundstate energy is given to a good approximation by the energy of the system of$N$ attractive bosons on an infinite line, provided the interaction is strongerthan the critical interaction strength of mean-field theory.Comment: 28 pages, 11 figure

Exact ground state of finite Bose-Einstein condensates on a ring