How is scattering possible? Bjorken and Drell's book shows that the 'in' and 'out' states are eigenstates of the full interacting theory. If this is true, then how is scattering possible if both in and out states are eigenstates of the full Hamiltonian of the interacting theory? Do I misunderstand something?
 A: That's because to compute the scattering matrix you don't evolve these "in" states with the interacting Hamiltonian alone (which would leave them unchanged) but with a mixture of the free and interacting Hamiltonians. Take a much simpler situation of a classical dynamical system (Rutherford scattering) for example a point charge coming for infinity and being scattered off a fixed charge with the same sign in a trajectory that looks like a hyperbola. Let $X$ be the set of states (i.e. position and velocity of the moving charge here).
The interacting evolution gives a flow $U(t_2,t_1):X\rightarrow X$ which means that $U(t_2,t_1)(x)$ is the state at time $t_2$ if $x$ was the state at time $t_1$. Note that I am not assuming $t_2>t_1$.
The flow satisfies the semigroup property
$$
U(t_3,t_1)=U(t_3,t_2)\circ U(t_2,t_1)\ .
$$
Likewise, if you remove the fixed charge then you get a free evolution flow $U_0(t_2,t_1)$. Clearly when time goes to $\pm\infty$ the moving charge is very far from the fixed one and so its evolution is approximately free, i.e.
$$
U(t_2,t_1)\simeq U_0(t_2,t_1)
$$
if $t_1,t_2$ are both sent to $-\infty$ (or to $+\infty$).
The purpose of the $S$ operator is to relate the asymptotically free evolution in the infinite past to that in the infinite future. 
The question is how do you label such asymptotes. a natural way is to use the position at time $0$. So given $x\in X$, the time zero data for a free evolution, you associate a free trajectory $t\mapsto U_0(t,0)(x)$. Using this labeling scheme how do you tell what is the future asymptote $y$ given that past asymptote $x$? The answer is
$$
y=U_0(0,T)\circ U(T,-T)\circ U_0(-T,0) (x)
$$
or rather the limit of that when $T\rightarrow\infty$. The $S$ matrix or operator is the map $x\rightarrow y$. As you can see it involves a mixture of free and interacting evolution operators. In the setting of classical ODEs this is basically the same idea as the method of variation of constants.
