38 research outputs found
Semigroup evolution in Wigner Weisskopf pole approximation with Markovian spectral coupling
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 (neglecting the background contribution) the evolution of a total
system subspace is not an exact semigroup for the multi-channel decay, unless
the projectors into eigesntates of the reduced evolution generator are
orthogonal. In this case these projectors must be evaluated at different pole
locations . Since the orthogonality relation does not
generally hold at different values of , for example, when there is symmetry
breaking, the semigroup evolution is a poor approximation for the multi-channel
decay, even for a very weak coupling. Nevertheless, there exists a possibility
not only to ensure the orthogonality of the projectors regardless the
number of the poles, but also to simultaneously suppress the effect of the
background contribution. This possibility arises when the theory is generalized
to take into account interactions with an environment. In this case , and
hence its eigenvectors as well, are {\it independent} of , which corresponds
to a structure of the coupling to the continuum spectrum associated with the
Markovian limit.Comment: 9 pages, 3 figure
Relaxation of metric constrained interpolation and a new lifting theorem
In this paper a new lifting interpolation problem is introduced and an explicit solution is given. The result includes the commutant lifting theorem as well as its generalizations in [27] and [2]. The main theorem yields explicit solutions to new natural variants of most of the metric constrained interpolation problems treated in [9]. It is also shown that via an infinite dimensional enlargement of the underlying geometric structure a solution of the new lifting problem can be obtained from the commutant lifting theorem. However, the new setup presented in this paper appears to be better suited to deal with interpolations problems from systems and control theory than the commutant lifting theorem