  A general hamiltonian $H$ of electrons in finite concentration, interacting
via any two-body coupling inside a crystal of arbitrary dimension, is
considered. For simplicity and without loss of generality, a one-band model is
used to account for the electron-crystal interaction. The electron motion is
described in the Hilbert space $S_\phi$, spanned by a basis of Slater
determinants of one-electron Bloch wave-functions. Electron pairs of total
momentum $K$ and projected spin $\zeta=0,\pm1$ are considered in this work. The
hamiltonian then reads $H=H_D+\sum_{K,\zeta}H_{K,\zeta}$, where $H_D$ consists
of the diagonal part of $H$ in the Slater determinant basis. $H_{K,\zeta}$
describes the off-diagonal part of the two-electron scattering process which
conserves $K$ and $\zeta$. This hamiltonian operates in a subspace of $S_\phi$,
where the Slater determinants consist of pairs characterised by the same $K$
and $\zeta$. It is shown that the whole set of eigensolutions $\psi, \epsilon$
of the time-independent Schr\"{o}dinger equation $(H-\epsilon)\psi=0$ divides
in two classes, $\psi_1,\epsilon_1$ and $\psi_2,\epsilon_2$. The eigensolutions
of class 1 are characterised by the property that for each solution
$\psi_1,\epsilon_1$ there is a single $K$ and $\zeta$ such that
$(H_D+H_{K,\zeta}-\epsilon_1)\psi_{K,\zeta}=0$ where in general $\psi_1 \ne
\psi_{K,\zeta}$, whereas each solution $\psi_2,\epsilon_2$ of class 2 fulfils
$(H_D-\epsilon_2)\psi_2=0$. We prove also that the eigenvectors of class 1 have
off-diagonal long-range order whereas those of class 2 do not. Finally our
result shows that off-diagonal long-range order is not a sufficient condition
for superconductivity.
