We investigate the spectrum of a model Hamiltonian with BCS and mean-field interaction in a finite domain under periodic boundary conditions. The model Hamiltonian is considered on the states of pairs and waves of density charges and their excitations. It is represented as the sum of three operators that describe noninteracting pairs, the interaction between pairs, and the interaction between pairs and waves of density charges. The last two operators tend to zero in the thermodynamic limit, and the spectrum of the model Hamiltonian coincides with the spectrum of noninteracting pairs with chemical potential shifted by mean-field interaction. It is shown that the model and approximating Hamiltonians coincide in the thermodynamic limit on their ground and excited states and both have two branches of eigenvalues and eigenvectors.