  Based on an exact treatment of hard-core bosons confined on one-dimensional
lattices, we obtain the large distance behavior of the one-particle density
matrix, and show how it determines the occupation of the lowest natural orbital
in the thermodynamic limit. We also study the occupation $\lambda_{\eta}$ of
the natural orbitals for large-$\eta$ at low densities. Both quantities show
universal behavior independently of the confining potential. Finite-size
corrections and the momentum distribution function for finite systems are also
analyzed.
