Exact Results for a Quantum Many-Body Problem in One Dimension

We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential V (r) =g{r^2} with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant g, as a determinant of order N. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.