SOME PROPERTIES OF THE EIGENSTATES IN THE MANY-ELECTRON PROBLEM

A general Hamiltonian H of electrons in finite concentration, interact ing via any two-body coupling inside a crystal of arbitrary dimension. is considered. For simplicity and without loss of generality, a one-b and 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. El ectron pairs of total momentum K and projected spin zeta=0,+/-1 are co nsidered in this work. The Hamiltonian then reads H=H-D+Sigma(K,zeta)H (K,zeta), where H-D consists of the diagonal part of H in the Slater d eterminant basis. H-K,H-zeta describes the off-diagonal part of the tw o-electron scattering process which conserves K and zeta. This Hamilto nian operates in a subspace of S-phi, where the Slater determinants co nsist of pairs characterized by the same K and zeta. It is shown that the whole set of eigensolutions psi,epsilon of the time-independent Sc hrodinger equation (H-epsilon)psi=0 divides into two classes, psi(1),e psilon(1) and psi(2),epsilon(2). The eigensoIutions of class 1 are cha racterized by the property that for each solution psi(1),epsilon(1) th ere is a single K and zeta such that (H-D + H-K,H-zeta-epsilon(1))psi( K,zeta)=0 where, in general, psi(1) not equal psi(K,zeta), whereas eac h solution psi(2),epsilon(2). of class 2 fulfills (H-D - epsilon(2))ps i(2)=0. We prove also that the eigenvectors of class I have off-diagon al long-range order, whereas those of class 2 do not. Finally, our res ult shows that off-diagonal long-range order is not a sufficient condi tion for superconductivity.