Consider the Lippmann-Schwinger equation (LSE)
$$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$
where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + i\eta}$.
If $|\phi\rangle$ is an incoming state, how do I determine the corresponding time-dependent incoming state? Is it simply $\hat{U}_0(t)|\phi\rangle$? Here $\hat{U}_0(t) = e^{-it\hat{H}_0}$.
If that is the case, then what is the time-dependent scattered state? Is it $\hat{U}_0(t)|\psi\rangle$? Why?
I also know that $\hat{G}_0(\epsilon)$ is the Fourier image of $\hat{G}_0(t)$ (see e.g. this Phys.SE post). Maybe I should somehow apply inverse Fourier transform to LSE to get time-dependent functions?
