Integral equations for scattering

In the previous sections the potential scattering problem has been defined in terms of a Schrodinger differential equation with outgoing spherical-wave boundary conditions. The description and computational methods are analogous to those used for one-electron bound-state problems. In this section we see that the whole problem in the coordinate representation can be written in terms of a single integral equation, which in many ways is easier to understand physically than the differential equation. 

A large breakthrough in physical transparency and ease of computation is achieved by expressing the problem as an integral equation in momentum space. The reason for this is that scattering experiments measure momenta, not positions, so that the momentum-space description parallels the experiment. 

The superscript (-I-) indicates outgoing spherical-wave boundary conditions. We will show that this corresponds to adding to a small, positive imaginary part, which will tend to zero. We multiply on the left by the inverse of the differential operator to obtain the formal solution 

The plane wave k) has been added to give the correct boundary condition. When V = 0, z (k)) = k). The inverse differential operator is the resolvent or free-particle Green s function operator and is denoted by Gq. 

We first replace the resolvent by a number by introducing its spectral representation and using the function theorem. At the same time we introduce the coordinate representation. 

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