Where does the \phi come from in the Lippmann-Schwinger equation?

[Unkraut]

This is probably a very stupid question as usual. I don't understand the Lippmann-Schwinger equation.

First we have the Schrödinger equation (H+V)\psi=E\psi, and we just rearrange it to \psi=\frac{1}{E-H}V\psi. But now, somehow magically this becomes \psi=\phi+\frac{1}{E-H}V\psi where \phi is a solution to H\phi=E\phi. Then we add a little imaginary quantity to the denominator just for the sake of being able to take an integral and let this imaginary quantity go to 0 in the end. This last step is not my problem. My question is: Where does the \phi come from?

 

[gabbagabbahey]

If you apply the operator E-H to both sides of each equation, |\psi\rangle=\frac{1}{E-H}V|\psi\rangle and |\psi\rangle=|\phi\rangle+\frac{1}{E-H}V|\psi\rangle, you get the same thing in each instance (since (E-H)|\phi\rangle=0 ), so it appears that either could be the correct solution.

However, you expect that |\psi\rangle\to|\phi\rangle in the limit V\to0 and only the second solution satisfies that condition.