by looking at the single-particle spectrum for odd particle numbers (with PBC) and for even particle numbers (with A-PBC) as a function of $q$, which is plotted in Fig. \[fig:oddevenspectrum\] for $M=4$ and $V_0 = 0.06 E_r$. Note that these numbers are chosen for visual simplicity, and the structure is exactly the same for $M=50$ and $V_0 = 0.02$ (with a slightly smaller gap). The structure of the two spectra reflects the difference in the average momentum, that is the spectrum for odd particle numbers close to $\Omega=2 \eta+1$ looks similar to the one for even particle numbers close to $\Omega=2 \eta$. The reason for this difference is that the ground-state is non-degenerate in the initial state for an odd number of particles, while it is two-fold degenerate in the initial state for an even number of particles. A lattice with an even number of sites therefore forces a splitting of the $M,M+1$ degenerate states for odd particle numbers, but there is no splitting for even particle numbers. An odd number of sites switches around the behavior for odd and even particle numbers, but the explanation is analogous. For deeper lattices we see that $\langle K \rangle/N$ grows linearly with 