Pairing correlations and collective $0^+$ states of nuclei. I

A Bose representation is obtained for the Hamiltonian of a microscopic model of a nucleus with residual pairing forces. This representation is used to show that the collective pairing excitation branches are weakly coupled to the noncollective branches. A collective Hamiltonian is constructed that describes pairing excitations. It is shown how one can pass from the collective Hamiltonian expressed in terms of boson operators to differential equations that are similar to the equations of the generalized model of a nucleus.