SINGLE-BETA AND DOUBLE-BETA DECAY FERMI TRANSITIONS IN AN EXACTLY SOLVABLE MODEL

An exactly solvable model suitable for the description of single- and double-beta decay processes of the Fermi type is introduced. The model is equivalent to the exact shell-model treatment of protons and neutr ons in a single-j shell. Exact eigenvalues and eigenvectors are compar ed to those corresponding to the Hamiltonian in the quasiparticle basi s (qp) and with the results of both the standard quasiparticle random phase approximation (QRPA) and the renormalized one (RQRPA). The role of the scattering term of the quasiparticle Hamiltonian is analyzed. T he presence of an exact eigenstate with zero energy is shown to be rel ated to the collapse of the QRPA. The RQRPA and the qp solutions do no t include this zero-energy eigenvalue in their spectra, probably due t o spurious correlations. The meaning of this result in terms of symmet ries is presented.