Projected basis for many-phonon systems

Abstract
The basis of a many-phonon system is constructed in the nuclear collective model using the projection-operator method. The proof of the completeness of this basis is given and the matrices of the generators of the group SO(5) are obtained in this basis. The matrix of the additional integral of motion Omega is calculated. This matrix has a non-degenerate spectrum of eigenvalues omega i which can be used as the missing quantum number.