Rigorous theorems on off-diagonal long-range order in the negative-UHubbard model

In this article, we study off-diagonal long-range order in the negative-U Hubbard model. We rigorously prove that all the elements of the two-particle reduced density matrix are positive, which may indicate a Bose condensation at q=0. Furthermore, we shall show that the ground state at half filling has pseudospin number 0. It implies that a sufficient condition proposed by Yang and Zhang [Int. J. Mod. Phys. B 5, 977 (1991)], for a state {Ψ} having off-diagonal long-range order, may not be a necessary condition.