Resonance S matrix, phase shift, and the cross section

Resonance S matrix, phase shift, and the cross section 

We consider the analytic continuation of the S(k) matrix into the complex k plane. Then, the phase shift is complex in general and the absolute value of 

S(k) can be of any positive value or zero. If S(k) has a pole at some complex k, the incoming-wave term is negligible there compared with the outgoing-wave term. If a pole lies at k = iic with a positive k, Eq. (12) behaves as gTKrg-iEt/h anc[ tpe potential supports a bound state (if k = iic) or what is called a virtual state (if k = —iK) of energy E = —h2K2/2m. 

The general form of S(E) satisfying the condition that S(E) = 1 for real E and having a pole, assumed as a simple pole, at E = Er — iT/2 = is expressible as 

The reduced energy e defined by Eq.(16) proves to be useful later. A quantity 

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