  An exactly solvable model is introduced, which is equivalent to the exact
shell-model treatment of protons and neutrons in a single j-shell for
Fermi-type excitations. Exact energies, quasiparticle numbers and double beta
decay Fermi amplitudes are computed and compared with the results of both the
standard quasiparticle random phase approximation (QRPA) and the renormalized
one (RQRPA), and also with those corresponding to the hamiltonian in the
quasiparticle basis (qp). A zero excitation energy state is found in the exact
case, occuring at a value of the residual particle-particle interaction at
which the QRPA collapse. The RQRPA and the qp solutions do not include this
zero-energy eigenvalue in their spectra, probably due to spurious correlations.
