Microscopic derivation of a collective bosonic Hamiltonian with the generalized density matrix method

The collective bosonic Hamiltonian is derived from the microscopic nucleonic Hamiltonian by the generalized density matrix method. Independent parameters in the collective Hamiltonian are fixed completely, solutions are given in detail. The random phase approximation corresponds to the harmonic potential of the current approach. The full solution (very close to the exact diagonalization) is obtained over the whole region of parameters including and beyond the instability point of the random phase approximation. The method is tested in the simple Lipkin model.