  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.

Quantum Physics;Superconductivity