  For a quantum system of N identical, harmonically interacting particles in a
one-dimensional harmonic trap we calculate for the bosonic and fermionic ground
state the corresponding 1-particle reduced density operator $\rho_1$
analytically. In case of bosons $\rho_1$ is a Gibbs state for an effective
harmonic oscillator. Hence the natural orbitals are Hermite functions and their
occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions
with not too large couplings the natural orbitals coincide up to just a very
small error with the bosonic ones. In case of strong coupling this still holds
qualitatively. Moreover, the decay of the decreasingly ordered fermionic
natural occupation numbers is given by the bosonic one, but modified by an
algebraic prefactor. Significant differences to bosons occur only for the
largest occupation numbers. After all the "discontinuity" at the "Fermi level"
decreases with increasing coupling strength but remains well pronounced even
for strong interaction.
