Comparison between wave functions in the random phase approximation, renormalized random phase approximation, and self-consistent random phase approximation methods

The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.