Estimate for regeneration up to the golden rule time

The authors study the regeneration contribution to the decay law in Wigner-Weisskopf theory for times less than and up to the golden rule time. A power series expansion for the regeneration term and the part of the product of the amplitudes which has the semigroup property is carried out in second-order perturbation theory, the same order to which the Wigner-Weisskopf calculation is carried out in their estimate of the line widths in atomic decay. They show that the regeneration contribution as a smaller leading behaviour in t than the amplitudes at times of the order of the golden rule time, thus accounting for an approximate semigroup behaviour, on this scale, within the framework of the Wigner-Weisskopf theory. For very short times, the estimates of Misra and Sinha (1977) are obtained.