Semigroup evolution in the Wigner-Weisskopf pole approximation with Markovian spectral coupling

Abstract

We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation the evolution of a total system subspace is not an exact semigroup for multichannel decay, unless the projectors into eigenstates of the reduced evolution generator W (z) are orthogonal. With multichannel decay, the projectors must be evaluated at different pole locations zα = z β , and since the orthogonality relation does not generally hold at different values of z, the semigroup evolution is a poor approximation for the multi-channel decay, even for very weak coupling. Nevertheless, if the theory is generalized to take into account interactions with an environment, one can ensure orthogonality of the W (z) projectors regardless the number of the poles. Such a possibility occurs when W (z), and hence its eigenvectors, are independent of z, which corresponds to the Markovian limit of the coupling to the continuum spectrum.