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We establish the relation between the Wigner-Weisskopf theory for thedescription of an unstable system and the theory of coupling to an environment.According to the Wigner-Weisskopf general approach, even within the poleapproximation (neglecting the background contribution) the evolution of a totalsystem subspace is not an exact semigroup for the multi-channel decay, unlessthe projectors into eigesntates of the reduced evolution generator $W(z)$ areorthogonal. In this case these projectors must be evaluated at different polelocations $z_\alpha\neq z_\beta$. Since the orthogonality relation does notgenerally hold at different values of $z$, for example, when there is symmetrybreaking, the semigroup evolution is a poor approximation for the multi-channeldecay, even for a very weak coupling. Nevertheless, there exists a possibilitynot only to ensure the orthogonality of the $W(z)$ projectors regardless thenumber of the poles, but also to simultaneously suppress the effect of thebackground contribution. This possibility arises when the theory is generalizedto take into account interactions with an environment. In this case $W(z)$, andhence its eigenvectors as well, are {\it independent} of $z$, which correspondsto a structure of the coupling to the continuum spectrum associated with theMarkovian limit.

Semigroup evolution in Wigner Weisskopf pole approximation with Markovian spectral coupling