Theory of Scattering Resonances

To understand how resonances affects the molecular phase, we briefly outline the basics of the theory of scattering resonances. We consider bound states ))s) interacting with a set of continuum states denoted E, n 1), where, as for the full scattering states , n-) [see Eq. (2.66)], the states E, n 1) approach the fine asymptotic solutions at infinite time  

Implicit in Eqs. (6.27) and (6.28) is that E, n 1) e P and ps) e Q, and as such they are orthogonal to one another. We, in fact, assume that each basis set spans the entire subspace to which it belongs, hence we can write an explicit representation of Q and P as 

For this expansion to be useful it is necessary that we have explicit expressions for ] ((f s E, n ) and for (E, n l i , n-). These are obtained below. 

Substituting this solution into Eq. (6.26) we obtain that 

An explicit representation of Eq. (6.36) is obtained by using the well-known identity (obtained by a similar contour integration to that depicted in Fig. 2.1), 

---