Solving real time evolution problems by constructing excitation operators

In this paper we study the time evolution of an observable in the interacting fermion systems driven out of equilibrium. We present a method for solving the Heisenberg equations of motion by constructing excitation operators which are defined as the operators \documentclass[12pt]{minimal}\begin{document}$\hat{A}$\end{document} satisfying \documentclass[12pt]{minimal}\begin{document}$[\hat{H},\hat{A}]=\lambda \hat{A}$\end{document}[Ĥ,Â]=λÂ. It is demonstrated how an excitation operator and its excitation energy λ can be calculated. By an appropriate supposition of the form of \documentclass[12pt]{minimal}\begin{document}$\hat{A}$\end{document} we turn the problem into the one of diagonalizing a series of matrices whose dimension depends linearly on the size of the system. We perform this method to calculate the evolution of the creation operator in a toy model Hamiltonian which is inspired by the Hubbard model and the nonequilibrium current through the single impurity Anderson model. This method is beyond the traditional perturbation theory in Keldysh-Green's function formalism, because the excitation energy λ is modified by the interaction and it will appear in the exponent in the function of time.