Microscopic structure and mathematical background of the multiconfigurational dynamical symmetry

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