  We consider nuclei composed of nucleons which interact via two-body
potentials decreasing exponentially at infinity. Protons and neutrons are not
distinguished in order to simplify notations. The basic result is the rigorous
mathematical proof that the bound eigenstates of the nuclei belong to the
subspace $P \mathscr{H}$ spanned by the states of a single hole in the ground
state $\psi_{0}$ of the parent nucleus with an extra nucleon, as in the
uncorrelated models. This follows from the exponential decay of $\psi_{0}$ when
a nucleon is very far apart from the residual nucleus. We prove that the real
difference from the uncorrelated models is that $P \mathscr{H}$ has an infinite
dimension and contains generalized single hole states, distinguishable from the
usual ones by the fact that one cannot assign a wave function to the hole. The
bound eigenstates of the nuclei are just states of this kind. Some physical
consequences are discussed, in particular the unexpected fact that the
dynamical part of the single-hole Hamiltonian, although nonnull, does not
affect the single-hole overlaps with the bound eigenstates. The decay of
$\psi_{0}$ provides the asymptotic behaviours of many single-hole quantities,
in particular the nuclear density matrix. Thus a by product of this paper is
the rigorous proof of the method developed by Van Neck, Waroquier and Heyde to
calculate overlaps and separation energies.
