Universal properties of hard-core bosons confined on one-dimensional   lattices
  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. 
Marcos Rigol and Alejandro Muramatsu