Exact memory integral in time-convolution generalized master equations: Argyres-Kelley projector

Recently, time integrals of memory functions from Time-Convolution Generalized Master Equations (TC-GME) with the Peier projector were calculated exactly. Similar procedure is applied in detail to the Argyres-Kelley projector yielding, in the Nakajima-Zwanzig projector formalism, TC-GME for the density matrix of the system. Specifying properly the meaning of the memory, the integral (if finite) is argued to yield exact zero in the interacting system-bath case provided that no approximations get involved and the thermodynamic limit of the bath is performed as usual before time-integration, This implies limitations on a simple-minded Born-Markov approximation often used in descriptions of kinetics of open systems and predicts slower-than-exponential relaxation to equilibrium in the long-time asymptotics. Origin of long-time tails of memories and reason for deviation from the usual correspondence with the Golden Rule are discussed.