Out-of-equilibrium Anderson model at high and low bias voltages

We study the high- and low-voltage properties of the out-of-equilibrium Anderson model for quantum dots, using a functional method in the Keldysh formalism. The Green's function at the impurity site can be regarded as a functional of a nonequilibrium distribution function. The dependence of the Green's function on the bias voltage V and temperature T arises through the nonequilibrium distribution function. From this behavior as a functional, it is shown that the nonequilibrium Green's function at high-voltage limit is identical to the equilibrium Green's function at high-temperature limit. This correspondence holds when the couplings of the dot and two leads, at the left and right, are equal. In the opposite limit, for small eV, the low-energy behavior of the Green's function can be described by the local Fermi-liquid theory up to terms of order $(eV)^2$. These results imply that the correlation effects due to the Coulomb interaction U can be treated adiabatically in the two limits, at high and low bias voltages.