The new method presented by B. Bremond and J. G. Valatin is generalized which at-tache ~ four body correlation to the Bogoliubov transformation. The generalization carried in this paper is similar to that from the Bogoliubov transformation to the generalized Bogoliubov one. We find a criterion of whether a trial ground state vector has an essential feature of four body correlation or it is like that of the generalized Bogoliubov transformation. This method is applied to a system with a charge independent pairing interaction. There appear two. different kinds of single quasiparticle excitation energy due to proton-neutron interaction.
We study variational wave functions of the product form, factorizing according to the wave vectors
k, for the ground state of a system of bosons interacting via positive pair interactions with a positive Fourier transform. Our trial functions are members of different orthonormal bases in Fock space. Each basis contains a quasiparticle vacuum state and states with an arbitrary finite number of quasiparticles. One of the bases is that of Valatin and Butler (VB), introduced fifty years ago and parametrized by an infinite set of variables determining Bogoliubov's canonical transformation for each
k. In another case, inspired by Nozi\`eres and Saint James the canonical transformation for
k=0 is replaced by a shift in the creation/annihilation operators. For the VB basis we prove that the lowest energy is obtained in a state with $\ensuremath{\sim}\sqrt{\text{volume}}$ quasiparticles in the zero mode. The number of
k=0 physical particles is of the order of the volume and its fluctuation is anomalously large, resulting in an excess energy. The same fluctuation is normal in the second type of optimized bases, the minimum energy is smaller and is attained in a vacuum state. Associated quasiparticle theories and questions about the gap in their spectrum are also discussed.