On the asymptotic condition The text that I am following is John Taylor's Scattering Theory. This relates specifically to page 28 and 29 where we discuss the asymptotic condition in quantum mechanics. What I am ultimately relating this to is the in/out states used in QFT to derive the LSZ-reduction formula, although this question does not relate specifically to quantum field theory.
The asymptotic condition is the following statement

If the potential $V$ satisfies some reasonable conditions regarding how quickly it drops off, for every $|\psi_{_{{\text{IN}}}}\rangle\in \mathcal H$ there exists a state $|\psi\rangle$ such that $$U(t)|\psi\rangle-U^0(t)|\psi_{_{{\text{IN}}}}\rangle=0 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{as}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t\rightarrow-\infty$$ and a similar condition holds for $|\psi_{_{{\text{OUT}}}}\rangle$ at $t\rightarrow +\infty$. In which $U(t)$ is the time evolution operator for the interacting Hamiltonian and $U^0(t)$ is the time evolution operator for the free Hamiltonian.

The claim is that if we take a state in the interacting theory and we consider it far enough back in time its time evolution is physically indistinguishable from that of the "in state" evolving under the free Hamiltonian. It basically makes sense of the long travel time of particles in scattering experiments relative to the time spent in the interaction with the potential.
I have a few questions about the implications of this condition which I will list below and I would greatly appreciate any answers to

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*Does the converse hold, that given any state evolving under the full interacting Hamiltonian in the asymptotic past, there exists another state whose time evolution in the asymptotic past under the free Hamiltonian is physically indistinguishable?


*For the above, is the correspondence 1-to-1 for a given (reasonable as defined above and in the book) potential?


*Is the space of all in (or separately, out) states isomorphic to the original Hilbert space $\mathcal H$? Or, in what you might consider sloppy physics words, we essentially have three identical Hilbert spaces $\mathcal H_\text{IN}$, $\mathcal H_\text{OUT}$ and $\mathcal H$?


*Is the scattering operator $\mathcal S$, which we think of as an operator $S:\mathcal H_{\text{IN}}\rightarrow \mathcal H_{\text{OUT}}$, an isomorphism between these space for a given (again, reasonable as defined above and in the book) potential?
 A: The book essentially goes on to answer these questions so I will just answer my question for anyone in future who is interested.


*

*Does the converse hold, that given any state evolving under the full interacting Hamiltonian in the asymptotic past, there exists another state whose time evolution in the asymptotic past under the free Hamiltonian is physically indistinguishable?


The answer is in general no as there can exist bound states which of course are not asymptotically free. In fact interestingly the space of bound states is orthogonal to the space of states with asymptotes. However in the case that there are no bound states then this does indeed hold.



*For the above, is the correspondence 1-to-1 for a given (reasonable as defined above and in the book) potential?


Yes, each in state defines exactly one regular state which identifies exactly one out state using the definition in the original question.



*Is the space of all in (or separately, out) states isomorphic to the original Hilbert space H? Or, in what you might consider sloppy physics words, we essentially have three identical Hilbert spaces $\mathcal H_\text{IN}$, $\mathcal H_\text{OUT}$ and $\mathcal H$?


Yes. It is an isomorphism despite being a subspace as it is infinite dimensional.



*Is the scattering operator S, which we think of as an operator S:HIN→HOUT, an isomorphism between these space for a given (again, reasonable as defined above and in the book) potential?


Yes.
