We present a general theory of non-Markovian dynamics for open quantum
systems. We explore the non-Markovian dynamics by connecting the exact master
equations with the non-equilibirum Green functions. Environmental back-actions
are fully taken into account. The non-Markovian dynamics consists of
non-exponential decays and dissipationless oscillations. Non-exponential decays
are induced by the discontinuity in the imaginary part of the self-energy
corrections. Dissipationless oscillations arise from band gaps or the finite
band structure of spectral densities. The exact analytic solutions for various
non-Markovian environments show that the non-Markovian dynamics can be largely
understood from the environmental-modified spectra of the open systems.Comment: 6 pages, 2 figure