  The Pauli exclusion principle is a constraint on the natural occupation
numbers of fermionic states. It has been suspected since at least the 1970's,
and only proved very recently, that there is a multitude of further constraints
on these numbers, generalizing the Pauli principle. Here, we provide the first
analytic analysis of the physical relevance of these constraints. We compute
the natural occupation numbers for the ground states of a family of interacting
fermions in a harmonic potential. Intriguingly, we find that the occupation
numbers are almost, but not exactly, pinned to the boundary of the allowed
region (quasi-pinned). The result suggests that the physics behind the
phenomenon is richer than previously appreciated. In particular, it shows that
for some models, the generalized Pauli constraints play a role for the ground
state, even though they do not limit the ground-state energy. Our findings
suggest a generalization of the Hartree-Fock approximation.
