Rigorous treatment of the damping theory for coupled systems in contact with reservoirs parametric amplifier

The damping theory is correctly applied to the coupled systems in contact with respective reservoirs. Differences between the present rigorous treatment of the damping theory and the conventional one are explicitly shown for an exactly solvable model of the parametric amplifier. The deviation becomes significant as the transition point of the system is approached. The conventional treatment breaks down for a description of a certain physical quantity even in the weak coupling limit.