What people mean by "state evolving with the interacting/free theory"?

Question

This is a quite basic question but I confess it is something I didn't get up to this point.

When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ evolving with the full interacting theory" and "states $\Psi_0$ evolving with the free theory". This is alluded to in this answer.

This paper also makes this clear:

Typically, one is interested in the overlap of scattering state, i.e., true eigenstates of the full Hamiltonian. Since these are usually unavailable, one resorts to descriptor states and an operator in such states - the $\cal{S}$ matrix - that describes scattering and can be expanded in a perturbative series.

But I'm surelly missing something here. I mean, given any dynamics $W(t)$ be it either $U(t)$ or $U_0(t)$ or any other, we can evolve any state with it.

I mean, we can pretty much consider $U_0(t)\Psi$ or $U(t)\Psi_0$. This makes me wonder what people precisely mean with states evolving with the interacting/free theory.

The guess, from the quotation of the paper, is that they mean "eigenstates of the full hamiltonian" and "eigenstates of the free hamiltonian".

But then there's something wrong in my understanding. I mean, if $\Psi$ is eigenstate of the full Hamiltonian, then

$$H\Psi=E\Psi$$

and hence the evolution $U(t)\Psi$ is trivial, it is just $\Psi(t)=e^{-iE t}\Psi$ and the overlap with any other eigenstate is zero.

So what people mean with "states evolving with the interacting theory" or "states evolving with the free theory"? If it is about the corresponding eigenstates, why isn't the evolution trivial as it seems?

Is this because the full Hamiltonian is time-dependent? But then, for potential scattering with a Coulomb potential for instance the Hamiltonian is time-independent $V(\mathbf{R})=g/|\mathbf{R}|$ and it seems the evolution would really be trivial.

I'm clearly missing something really basic here. What is it? In summary:

1. Why people talk about "states evolving with free/interacting theory"? Can't we use any state as initial condition for any evolution $W(t)$ be it free/interacting?

2. How these states are characterized? Are they eigenstates of the free/interacting Hamiltonian? If so, why their evolution isn't trivial as outlined in the question?

Answer 1

In scattering theory, one would like to approximate, when time is large, a complicated interacting evolution with something simpler, i.e. the free evolution.

The basic physical justification goes as follows. If an interaction is suitably localized in space, and if one considers configuration that "escape at infinity", i.e. that as time goes by get very far from the region where the interaction is going on, then such configurations will asymptotically behave as a free theory but with an initial datum that is in general different from the original one.

Mathematically speaking, let $U(t)$ be the interacting evolution, and $U_0(t)$ the free one. The aim is to prove that for all (or almost all) $\psi$, there exists $\psi_{\pm}$ (the asymptotic state) such that $$\lim_{t\to\pm\infty}\lVert U(t)\psi - U_0(t)\psi_{\pm}\rVert=0\; .$$ If that is true, then the complicated system described by $U(t)\psi$ can be described, to a very good approximation if one waits enough time in the future or past, by the simpler evolution $U_0(t)\psi_{\pm}$. In this case we say that the state $\psi$ scatters, with corresponding asymptotic states $\psi_{\pm}$.

Since both $U_0(t)$ and $U(t)$ are unitary operators, the above is equivalent to saying that $$\lim_{t\to\pm\infty}\lVert U_0(t)^*U(t)\psi - \psi_{\pm}\rVert=0\; .$$ In other words, this amounts to studying the limit, in the strong topology, of $U_0(t)^*U(t)$ (and also, conversely, of $U(t)^*U_0(t)$).

There are, however, states for which one immediately sees that such convergence is impossible. Let $\psi_\lambda$ be an eigenvector of $U(t)$: $U(t)\psi_\lambda=e^{-it\lambda}\psi_\lambda$. Then $U_0(t)^*U(t)\psi_\lambda=e^{it(H_0-\lambda)}\psi_\lambda$, and such vector has strong limit as $t\to\pm\infty$ if and only if $\psi_\lambda$ is also an eigenvector of $H_0$ with eigenvalue $\lambda$. However, it is physically implausible that the interacting and free theories that are being compared share common eigenvalues and eigenvectors, and therefore one should get rid of such eigenvectors of $U(t)$, in order to prove the strong convergence of $U_0(t)^*U(t)$ (since these states clearly do not scatter). This is usually done projecting out of the pure point spectrum of $U(t)$. Physically, this also means that for Hamiltonians with pure point spectrum no state scatters, and this is because in this case the states cannot escape the region where there is interaction (Hamiltonians with pure point spectrum describe trapped systems). The converse is also true for $U^*(t)U_0(t)$, but usually the free reference dynamics $U_0(t)$ is chosen in such a way that it has purely continuous spectrum (e.g. the free particle Hamiltonian), and therefore it is not necessary to project out of the pure point spectrum, since the latter is empty.