Scattering theory Lippmann-Schwinger

In a true scattering problem, an incident wave is specified, and scattered wave components of ifr are varied. In MST or KKR theory, the fixed term x in the full Lippmann-Schwinger equation, f = x + / GqVms required to vanish, x is a solution of the Helmholtz equation. In each local atomic cell r of a space-filling cellular model, any variation of i// in the orbital Hilbert space induces an infinitesimal variation of the KR functional of the form 8 A = fr Govi/s) + he. This... 

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. 

Our interest in Brillouin-Wigner perturbation theory was stimulated by our finding [44] that this theoretical tool proved very useful in the scattering theory. The fundamental equation, known in the scattering theory as Lippmann-Schwinger equation, expresses the scattering operator as... 

In contrast to the MP theory, interaction V in the Lippmann-Schwinger equation is strong and cannot be considered as a small perturbation. Hence, the scattering amplitude... 

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). 

The title of this subsection refers to a recent paper by Kylstra and Joachain investigating double poles of the S matrix [25]. Their paper is based on the time-dependent Lippmann-Schwinger equation. Since the Hamiltonian is periodic in time the use of the Floquet theorem permits one to apply the time-independent theory of scattering. As in Ref. [25] we consider two quasi bound states (n = 2). Instead of starting from an Hamiltonian periodic in time (semi-classical approximation) we use a time-independent model. The laser field is assumed to be quantized and as a result the total Hamiltonian describing the atom in the laser field is time-independent (see chapter VI of Ref. [13]) Our aim is twofold To reproduce the results of Ref. [25] and, more generally, to illustrate the relevance of simple models to describe collision... 

The above procedure resembles the Lippmann-Schwinger method [106] for constructing the incoming state in scattering theory. 

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