S matrix for an isolated resonance

A QBS formed in some way or other may decay into any channel i of the same symmetry as O at a rate Ift/ft that may differ depending on the channel. The element Sy (i,j = 1,2. N0) must include both T, and r) on an equal footing because of the time-reversal symmetry. The lifetime r of state i is the inverse of the rate T /ft of decay into all channels, i.e., the sum of all I ,/ft, the total width r being T.  

We define a normalized column vector g and a projection matrix P by 

However, the assumption of no inelastic scattering off resonance is too restrictive and often unrealistic. Hence, we consider a more general form in the following, which we call the Breit-Wigner S matrix in this article, namely [8,35, 37], 

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