Rigorous bounds on the susceptibilities of the Hubbard model

Rigorous bounds on the susceptibilities of the single-band Hubbard model which hold in all dimensions are presented. In the attractive model the spin susceptibility is bounded above by (4‖U‖${)}^{\mathrm{-}1}$ where U(<0) is the on-site interaction potential. In the half-filled repulsive model on a bipartite lattice the charge and the on-site pairing susceptibilities are bounded above by ${\mathit{U}}^{\mathrm{-}1}$. The present result implies that the susceptibilities never diverge in the above mentioned circumstances and also the absence of corresponding long-range order.