Nonstationary perturbation theory for the energy shifts of a degenerate level

The asymptotic (when $T\equiv t_1-t_2\to\infty$ ) representation for the operator $PS(t_1,t_2)P$ where $P$ is the projector on some degenerate subspace of the nonperturbed energy level and $S(t_1,t_2)$ is the operator of the time development in the interaction picture is obtained. The asymptotic formula is the following:
$$PS(t_1,t_2)P=R_0\exp (-iQT)=(\exp\{-iQ^+T\})R_0=R_0^{1/2}(\exp\{-i\bar QT\})R_0^{1/2},$$
where $Q$ is the nonhermitian secular operator [3], $R_0$ and $\bar Q$ are the hermitian operators.