Microscopic structure and mathematical background of the multiconfigurational dynamical symmetry

Background: The multiconfigurational dynamical symmetry (MUSY) is the common intersection of the shell, collective, and cluster models for the multi-major-shell problem. It is able to describe the spectra of different configurations in different energy windows in a unified way. It is based on some heuristic arguments, related to the connection of the wave functions, and energy eigenvalues. The detailed mathematical background has been worked out so far only for the simplest case of the two binary cluster configurations.Purpose: I intend to construct the exact algebraic framework for the general case of the MUSY, i.e., for any number of configurations and any number of clusters or nucleons. As an illustrative example, the spectrum of different configurations of the $^{44}\mathrm{Ti}$ nucleus is described by a simple Hamiltonian.Methods: Classification schemes defined by different algebra chains need to be combined; in particular, those of the major shell scheme and of the particle index scheme.Results: A class of Hamiltonians, which is invariant under the transformations from one configuration to the other, is determined. In case of the $^{44}\mathrm{Ti}$ low-lying shell-model spectrum, as well as $^{40}\mathrm{Ca}+^{4}\mathrm{He}$ and $^{28}\mathrm{Si}+^{16}\mathrm{O}$ cluster states are obtained in a unified way.Conclusions: The MUSY is based on two pillars: (i) a unified multiplet structure for shell, collective, and cluster model states and (ii) a Hamiltonian which is invariant with respect to the transformations from one configuration to the other.