Validity of the BCS and RPA approximations in the pairing-plus-monopole solvable model

Abstract

A solvable many-body model, which simulates qualitatively the main characteristics of the pairing-plus-quadrupole model, is presented and investigated for systems of even and odd numbers of particles. The exact solution, when the force parameters are fixed and the number of particles is varied throughout the “major shell” of the model, is compared to the approximate BCS (or HB) solution for the ground state, and to the quasi-particle RPA result for the one-phonon excited states. We conclude that the BCS (or HB) solution for the ground state energy and wave function is good and practically exact if the number-of-particles projection is performed. The RPA for even systems seems to work for the so-called “pairing vibration” (0 +), whereas, for the model “quadrupole vibration”, errors of about 20% of the exact excitation are encountered. For odd systems, we examine a particle-conserving approximation for the corresponding quadrupole vibration.