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.Comment: 48 pages, 28 figures, 6 table
