Abstract
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 neutrons in a single-$j$ shell. Exact eigenvalues and eigenvectors are compared to those corresponding to the Hamiltonian in the quasiparticle basis (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. The presence of an exact eigenstate with zero energy is shown to be related to the collapse of the QRPA. The RQRPA and the qp solutions do not include this zero-energy eigenvalue in their spectra, probably due to spurious correlations. The meaning of this result in terms of symmetries is presented.
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