Lippmann-Schwinger solution What's wrong with this general solution of the Lippmann-Schwinger equation:
$$
|\psi_k \rangle=|\phi_k \rangle+G_k V|\psi_k \rangle 
$$
Taking the inner product with $\langle\phi_{k'}|$
\begin{align}
\langle \phi_{k'}|\psi_k \rangle&=\langle \phi_{k'} |\phi_k \rangle+\langle \phi_{k'} |G_k V|\psi_k \rangle \\
&=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |V|\psi_k \rangle\\
&=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |H-H_0|\psi_k \rangle\\
&=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |E_k-E_{k'}|\psi_k \rangle
\end{align}
Basic algebra then gives us
$$\langle \phi_{k'}|\psi_k \rangle[1-g(E_k-E_{k'})(E_k-E_{k'})]=\delta(k'-k)$$
and isolating the inner product on the left hand side gives us
$$\langle \phi_{k'}|\psi_k \rangle=\delta(k'-k) [1-g(E_k-E_{k'})(E_k-E_{k'})]^{-1}$$
where $g(E_k-E_{k'})$ is the eigenvalue of $G_k$ resulting from hitting $\langle \phi_{k'} |$ on the left?
This seems to suggest that the eigenstate $|\psi_k \rangle $ of the full Hamiltonian $H$ is the same as the eigenstate  $|\phi_k \rangle$ of $H_0$, up to a proportionality factor, since $|\psi_k \rangle $ has no amplitude to be in $|\phi_{k'\neq k} \rangle$.  
 A: It seems to me that the passage from the 4th line to the 5th cannot be right because $[E_k]_k$ stands for the spectrum of the free hamiltonian $H_0$. And what you've done here is :
$$\langle \phi_k|H=E_k \langle \phi_k|
$$
which is wrong. $[E_k-E_{k'}]_{(k,k')}$ cannot be the spectrum of the potential $V$.
If you want the solution of this equation, you must compute an expansion on the Green operator $\hat{G}$ following the Dyson equation :
$$\hat{G}=\hat{G_0}+\hat{G_0}\hat{V}\hat{G}$$
with $\hat{G_0}$ the free Green operator associated to $H_0$ ; and stop it at the order in $V$ you feel your calcultaions will be consistent. 
For instance, at first order :
$$\hat{G}=\hat{G_0}+\hat{G_0}\hat{V}\hat{G_0}$$ 
i.e. with your notations : $|\psi_k\rangle=|\phi_k\rangle+\hat{G}_k\hat{V}|\phi_k\rangle$.
A: $\langle \phi_{k'} \rvert$ is not an eigenfunction of $G_k$, hence the second line of your derivation is not correct. The eigenfunctions are $\langle \psi_{k} \rvert$, and that is exactly what we usually solve Lippmann-Schwinger equation for.
