The N-representability problem, the pseudo-spectral decomposition of antisymmetric 1-body operators, and collective behaviour

The pseudo--spectral decomposition of an $N$--particle antisymmetric 1--body positive--semidefinite operator that corresponds to the canonical convex decomposition into the extreme elements of the dual cone of the set of fermion $N$--representable $1$--density operators has been derived. An attempt at constucting a mathematical model for collective behaviour of a system of $N$--fermions that originates from the pseudo--spectral decomposition is presented.