  The non-vanishing of the natural orbital occupation numbers of the
one-particle density matrix of many-body systems has important consequences for
the existence of a density matrix-potential mapping for nonlocal potentials in
reduced density matrix functional theory and for the validity of the extended
Koopmans' Theorem. On the basis of Weyl's theorem we give a connection between
the differentiability properties of the ground state wave function and the rate
at which the natural occupations approach zero when ordered as a descending
series. We show, in particular, that the presence of a Coulomb cusp in the wave
function leads, in general, to a power law decay of the natural occupations,
whereas infinitely differentiable wave-functions typically have natural
occupations that decay exponentially. We analyze for a number of explicit
examples of two-particle systems that in case the wave function is non-analytic
at its spatial diagonal (for instance, due to the presence of a Coulomb cusp)
the natural orbital occupations are non-vanishing. We further derive a more
general criterium for the non-vanishing of NO occupations for two-particle wave
functions with a certain separability structure. On the basis of this criterium
we show that for a two-particle system of harmonically confined electrons with
a Coulombic interaction (the so-called Hookium) the natural orbital occupations
never vanish.
