I am currently reading through Weinsteins Lectures on Quantum Mechanics. In Chapter 8 he deals with general scattering theory.
Let
$$H = H_0 + V(\vec{x})$$
where $V(\vec{x})$ is a central potential, that satisfies $V(\vec{x}) \rightarrow 0$ for $\vert \vec{x} \vert \rightarrow \infty$. In addition to that, we introduce $\Psi^{\pm}_{\alpha}$ (interpreted as in- and out-states) as eigenstates of the full Hamiltonian
$$H\Psi^{\pm}_{\alpha} = E_{\alpha}\Psi^{\pm}_{\alpha},$$
as well as $\Phi_{\alpha}$, which are defined as eigenstates of the free-particle Hamiltonian
$$H_0 \Phi_{\alpha} = E_{\alpha}\Phi_{\alpha}$$
With that being said, the following causes some trouble to me:
The definition of $\Psi_{\alpha}^{+}$ and $\Psi_{\alpha}^{-}$ can be made more precise by specifying that if $g(\alpha)$ is a sufficiently smooth function of the momenta in the state $\alpha$, then
$$\int d\alpha \, g(\alpha) \,\Psi^{\pm}_{\alpha}\,\exp\left(-\frac{iE_{\alpha}t}{\hbar}\right) \rightarrow \int d\alpha \, g(\alpha) \,\Phi_{\alpha} \, \exp\left(-\frac{iE_{\alpha}t}{\hbar}\right)$$
for $t \rightarrow \mp \infty$.
I am not really sure, what to make out of this. I think, that Weinberg wants to emphasise, that the in- and out-states $\Psi^{\pm}_{\alpha}$ look like states of the free-particle Hamiltonian, if the scattering event either will take place in the (far) distant future or did already happen in the long-gone past. However in this case, I would have either expected
$$\int d\alpha \, g(\alpha) \,\Psi^{\pm}_{\alpha}\,\exp\left(-\frac{iE_{\alpha}t}{\hbar}\right) \rightarrow \int d\alpha \, g(\alpha) \,\Phi_{\alpha}$$
for $t \rightarrow \mp \infty$ or something like
$$\int d\alpha \, g(\alpha) \,\Psi^{\pm}_{\alpha,t_0}\,\exp\left(-\frac{iE_{\alpha}t}{\hbar}\right) \rightarrow \int d\alpha \, g(\alpha) \,\Phi_{\alpha} \, \exp\left(-\frac{iE_{\alpha}t}{\hbar}\right)$$
for $t_0 \rightarrow \mp \infty$ ($t_0$ marks the starting point of our observation)
An explanation of why the expression from Weinstein makes sense, would be much appreciated.