A soluble model for time dependent perturbations of an unstable quantum system

We show that a time dependent addition to the coupling constant of the Lee-Friedrichs model results in an exactly soluble model for the effect of time dependent perturbations of an unstable quantum system. For a perturbation of a single frequency, a shift in the position of the original Lee-Friedrichs pole, and the emergence of a harmonically associated sequence of other poles, are obtained by iteration of the exact difference equations for the reduced resolvent. In lowest order, line broadening can be seen directly in the structure of the shifted pole position. If the time dependence contains modulation, e.g., with two commensurate frequencies, we show that there are contributions which could lead to line narrowing as well.