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.