Asymptotics of the nucleus ground-state and single-hole nature of the bound states of the nuclei

Abstract

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.