We apply self-consistent second order perturbation theory (SCSOPT) with respect to the on-site repulsive interaction U to study the Hubbard model in two dimensions. We investigate single particle properties of the model over the entire doping range at zero temperature. It is shown that as doping decreases toward half-filling
ω-mass enhancement factor increases, while k-mass enhancement factor decreases. The increase in
ω-mass enhancement factor is larger than the decrease in k-mass enhancement factor, so that total-mass is larger than that in the non-interacting case. When particle number density per unit cell n is given by 0.64
Dynamical response functions of the Hubbard model on an infinite dimensional lattice (D=\ensuremath{\infty}) are investigated by the self-consistent second-order perturbation theory with respect to the Coulomb interaction U. The vertex corrections are properly considered so that the conservation laws are satisfied, i.e., the f-sum rule is satisfied. The explicit expression for the f-sum rule is obtained in the case of infinite dimensions. It is found that even in the second-order perturbation theory the electronic states show many-body features at finite temperature and/or frequency regions: the system crosses over from the Fermi-liquid state at the low temperature and low-frequency region to the incoherent regime at high temperatures.