Abstract
We give a variational foundation for the improper Hartree-Bogoliubov (HE) formalism by which odd fermion systems can be treated in an analogous way as the formalism of even fermion systems. Because of the difference of the normalization for the improper HE amplitudes from that for the proper ones, the improper HE equation cannot be diagonalized with the conventional BCS ansatz. We decompose the improper liB amplitudes into the amplitudes describing paired and unpaired modes of fermions and the variational procedure is applied to them. We obtain a new form of the improper HE equation with respect to the paired mode amplitudes. It includes the effects of unpaired modes but can be diago nalized with the conventional manner.
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