Microscopic multiphonon approach to nuclei with a valence hole in the oxygen region

An equation of motion phonon method, developed for even nuclei and recently extended to odd systems with a valence particle, is formulated in the hole-phonon coupling scheme and applied to $A=15$ and $A=21$ isobars with a valence hole. The method derives a set of equations which yield an orthonormal basis of states composed of a hole coupled to an orthonormal basis of correlated $n$-phonon states ($n=0,1,2,\ensuremath{\cdots}$), built of constituent Tamm-Dancoff phonons, describing the excitations of a doubly magic core. The basis is then adopted to solve the full eigenvalue problem. The method is formally exact but lends itself naturally to simplifying approximations. Self-consistent calculations using a chiral Hamiltonian in a space encompassing up to two-phonon and three-phonon basis states in $A=21$ and $A=15$ nuclei, respectively, yield full spectra, moments, electromagnetic and $\ensuremath{\beta}$-decay transition strengths, and electric dipole cross sections. The analysis of the hole-phonon composition of the eigenfunctions contributes to clarify the mechanism of excitation of levels and resonances and to understand the reasons of the deviations of the theory from the experiments. Prescriptions for reducing these discrepancies are suggested.