Collective particle hole excitation in a deformed ground state

It is well known that the Hartree-fock ground state of an infinite system cannot support collective particle hole excitations for an attractive residual interaction. On the other hand, the break down of the random phase approximation (RPA) for attractive interactionsis believed to signal the onset of a phase transition which is associated with a broken symmetry of the ground state. In the case of an infinite system, the question naturally arises as to whether a nonuniform ground state which breaks translational symmetry exists, and whether this new ground state can support collective quasi-particle quasi-hole excitations in terms of a suitably chosen single particle basis. It has been found that a self-consistent non-uniform density distribution does give rise to a lower total energy of the ground state. In the article the possibility and the properties of collective density oscillations in a non-uniform medium are investigated. From the study it appears, that, while Hartree-fock and RPA are not reliable in the vicinity of the interaction strength for which the phase transition occurs, it can be reliably used beyond the interaction strength