Two Theorems on the Hubbard-Hirsch Model

In this paper, the two theorems on the Hubbard-Hirsch model are shown rigorously. For positive on-site interaction U, the model on some bipartite lattices has a weak ferromagnetic ground state at half filling. For negative U, the ground state of the model on some bipartite lattices possesses off-diagonal long-range order within a finite range of the electron filling factor.