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Some properties of the eigenstates in the many-electron problem

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 $S_ϕ$, spanned by a basis of Slater determinants of one-electron Bloch wave-functions. Electron pairs of total momentum $K$ and projected spin $ζ=0,\pm1$ are considered in this work. The hamiltonian then reads $H=H_D+\sum_{K,ζ}H_{K,ζ}$, where $H_D$ consists of the diagonal part of $H$ in the Slater determinant basis. $H_{K,ζ}$ describes the off-diagonal part of the two-electron scattering process which conserves $K$ and $ζ$. This hamiltonian operates in a subspace of $S_ϕ$, where the Slater determinants consist of pairs characterised by the same $K$ and $ζ$. It is shown that the whole set of eigensolutions $ψ, ε$ of the time-independent Schrödinger equation $(H-ε)ψ=0$ divides in two classes, $ψ_1,ε_1$ and $ψ_2,ε_2$. The eigensolutions of class 1 are characterised by the property that for each solution $ψ_1,ε_1$ there is a single $K$ and $ζ$ such that $(H_D+H_{K,ζ}-ε_1)ψ_{K,ζ}=0$ where in general $ψ_1 \ne ψ_{K,ζ}$, whereas each solution $ψ_2,ε_2$ of class 2 fulfils $(H_D-ε_2)ψ_2=0$. 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.