The distorted-wave transformation

It would be convenient for solving the Lippmann—Schwinger equation (6.73) if we could make the potential matrix elements as small as possible. For example, we could hope to find a transformed equation whose iteration would converge much more quickly. This is achieved by a judicious choice of a local, central potential U, which is called the distorting potential since the problem is reformulated in terms of the distorted-wave eigenstates of U rather than the plane waves of (6.73). An important particular case of U is the Coulomb potential Vc in the case where the target is charged. The Hamiltonian (6.2) is repartitioned as follows 

By multiplying (6.70) on the left by the channel Schrodinger operator — K we obtain the form of the Schrodinger equation that embodies the correct collision boundary conditions. Using (6.76) this becomes 

The solution of (6.78) is the distorted-wave channel state. Its more-explicit form is written in analogy to equn. (6.7) for the channel state, with the distorted waves k - ) replacing the plane waves kj). It is convenient 

The transformed integral equation, corresponding to (6.77), is given for physical boundary conditions indicated by the superscript (-1-). 

We may arrive at this equation by repeating the derivation of this chapter for the Hamiltonian (6.76). 

---