Lippmann-Schwinger formalism

This equation is similar to the Lippmann-Schwinger equation extended in the complex plane (see Chapters 2.1. and 2.5. of Ref. [11]). However, there is a fundamental difference between our approach and the Lippmann-Schwinger formalism. Here Cl(z) establishes a one-to-one correspondence between a finite number of quasi-bound states belonging to the model space (projector Pq) and the same number of states belonging to the complementary space (projector Qo) whereas the Mpller wave operator establishes... 

Use of the potential (7.35) in solving the coupled Lippmann—Schwinger equations (6.73,6.87) corresponding to (7.24) is a unique and numerically-valid description of the electron—atom scattering problem in the context of formal scattering theory. 

The Lippmann—Schwinger equations (6.73) are written formally in terms of a discrete notation i) for the complete set of target states, which includes the ionisation continuum. For a numerical solution it is necessary to have a finite set of coupled integral equations. We formulate the coupled-channels-optical equations that describe reactions in a channel subspace, called P space. This is projected from the chaimel space by an operator P that includes only a finite set of target states. The entrance channel 0ko) is included in P space. The method was first discussed by Feshbach (1962). Its application to the momentum-space formulation of electron—atom scattering was introduced by McCarthy and Stelbovics... 

The coupled-channels-optical equations are formally analogous to the Lippmann—Schwinger equivalent of (7.29) in which the coupling potential includes the potential V (7.40) and a polarisation potential that describes the real (on-shell) and virtual (off-shell) excitation of the complementary channel space, called Q space. The total coupling potential is the optical potential... 

We return now to the coupled-channels approach based on operator equations. The formalism is adequately covered in several books (see e.g. Goldberger and Watson, 1964 Newton, 1966 Levine, 1969) and we shall only present the main equations. Assume for simplicity that only two reaction channels a (for A + BC)and 6(AB + C) exist. The total Hamiltonian H may be split into two terms, a channel Hamiltonian Hc for free motion and a channel interaction Vc, with c a, b. If a is the initial free state and we want the scattering states in channel a, i.e. those in the absence of rearrangement, then the Lippmann-Schwinger equation gives... 

Here the second of these expressions is obtained by introducing the transition operator (51). Equation (59) is the Lippmann-Schwinger equation of formal scattering theory, which describe how each monochromatic component of the incoming wave packet is distorted by the scattering interaction (Levine, 1969). 

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