Semigroup evolution in the wigner-weisskopf pole approximation with markovian spectral coupling

F. Shikerman, A. Peer, L. P. Horwitz

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3 Scopus citations


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≠ zβ, and since the orthogonality relation does not generally hold at different values of z, the semigroup evolution is a poor approximation for the multichannel 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 of the number of poles. Such a possibility occurs when W(z), and hence its eigenvectors, is independent of z, which corresponds to the Markovian limit of the coupling to the continuum spectrum.

Original languageEnglish
Article number012122
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Issue number1
StatePublished - 27 Jul 2011


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