The Bohr hamiltonian, also called collective hamiltonian, is one of the cornerstone of nuclear physics and a wealth of solutions (analytic or approximated) of the associated eigenvalue equation have been proposed over more than half a century (confining ourselves to the quadrupole degree of freedom). Each particular solution is associated with a peculiar form for the V (β, γ) potential. The large number and the different details of the mathematical derivation of these solutions, as well as their increased and renewed importance for nuclear structure and spectroscopy, demand a thorough discussion. It is the aim of the present monograph to present in detail all the known solutions in γ−unstable and γ−stable cases, in a taxonomic and didactical way. In pursuing this task we especially stressed the mathematical side leaving the discussion of the physics to already published comprehensive material.The paper contains also a new approximate solution for the linear potential, and a new solution for prolate and oblate soft axial rotors, as well as some new formulae and comments, and an appendix on the analysis of a few interesting numerical sequences appearing in this context. The quasi-dynamical SO(2) symmetry is proposed in connection with the labeling of bands in triaxial nuclei.

Background: The 29 F system is located at the lower-N boundary of the "island of inversion" and is an exotic, weakly bound system. Little is known about this system beyond its two-neutron separation energy (S2n) with large uncertainties. A similar situation is found for the low-lying spectrum of its unbound binary subsystem 28 F.Purpose: To investigate the configuration mixing, matter radius and neutron-neutron correlations in the groundstate of 29 F within a three-body model, exploring the possibility of 29 F to be a two-neutron halo nucleus. Method:The 29 F ground-state wave function is built within the hyperspherical formalism by using an analytical transformed harmonic oscillator basis. The Gogny-Pires-Tourreil (GPT) nn interaction with central, spin-orbit and tensor terms is employed in the present calculations, together with different core + n potentials constrained by the available experimental information on 28 F. Results:The 29 F ground-state configuration mixing and its matter radius are computed for different choices of the 28 F structure and S2n value. The admixture of d-waves with pf components are found to play an important role, favouring the dominance of dineutron configurations in the wave function. Our computed radii show a mild sensitivity to the 27 F + n potential and S2n values. The relative increase of the matter radius with respect to the 27 F core lies in the range 0.1 -0.4 fm depending upon these choices.Conclusions: Our three-body results for 29 F indicate the presence of a moderate halo structure in its ground state, which is enhanced by larger intruder components. This finding calls for an experimental confirmation.

Giant resonances are collective excitation modes for many-body systems of fermions governed by a mean field, such as the atomic nuclei. The microscopic origin of such modes is the coherence among elementary particle-hole excitations, where a particle is promoted from an occupied state below the Fermi level (hole) to an empty one above the Fermi level (particle). The same coherence is also predicted for the particle–particle and the hole–hole excitations, because of the basic quantum symmetry between particles and holes. In nuclear physics, the giant modes have been widely reported for the particle–hole sector but, despite several attempts, there is no precedent in the particle–particle and hole–hole ones, thus making questionable the aforementioned symmetry assumption. Here we provide experimental indications of the Giant Pairing Vibration, which is the leading particle–particle giant mode. An immediate implication of it is the validation of the particle–hole symmetry.