The following is from Talagrand's What is a Quantum Field Theory (Section 12.2). He lets $H$ be the Hamiltonian $$H = H_0 + V$$ where $H_0$ is the Hamiltonian of the free particle and $V$ is some potential. He also lets $$U_0(t) = \exp(-tH_0), \quad U(t) = \exp(-itH)$$ be unitary operators governing their respective time evolutions. He then says (Proposition 12.2.6)

Assume that $$\|V\|^2 := \int d^3x V(x)^2 < \infty.$$ Then for each $\varphi$ in the state space $H$, the limits $$\Omega_+(\varphi) = \lim_{t \rightarrow -\infty} U(t)^{-1}U_0(t)(\varphi)\\ \Omega_-(\varphi) = \lim_{t\rightarrow \infty} U(t)^{-1}U_0(t)(\varphi)$$ exist.

I am having some trouble interpreting what the operators $\Omega_-$ and $\Omega_+$ represent. Talagrand provides some motivation in the pages before the proposition, and says that

$\Omega_+(x,v) = (x,v)_{\text{in}}$ represents the state at time $t = 0$ of the particle whose motion in the far past resembled the motion of a free particle of position and velocity $(x,v)$ at time $0$.

One defines similarly an operator $\Omega_-$ by starting with the free particle of position $x$ and velocity $v$ at time zero, running time forward until a large positive time $t_0$, turning the potential on, running time backward until we reach time zero again, and taking the limit as $t_0 \rightarrow \infty$. One also uses the notation $(x,v)_{\text{out}} = \Omega_-(x,v)$.

However I still am a little lost on what these $\Omega$ operators are doing exactly, and what is the physical reasoning behind defining them as well as the scattering operator $S = \Omega_-^{-1}\Omega_+$.

The following is from Talagrand's What is a Quantum Field Theory (Section 12.2). He lets $H$ be the Hamiltonian $$H = H_0 + V$$ where $H_0$ is the Hamiltonian of the free particle and $V$ is some potential. He also lets $$U_0(t) = \exp(-itH_0), \quad U(t) = \exp(-itH)$$ be unitary operators governing their respective time evolutions. He then says (Proposition 12.2.6)

Assume that $$\|V\|^2 := \int d^3x V(x)^2 < \infty.$$ Then for each $\varphi$ in the state space $H$, the limits $$\Omega_+(\varphi) = \lim_{t \rightarrow -\infty} U(t)^{-1}U_0(t)(\varphi)\\ \Omega_-(\varphi) = \lim_{t\rightarrow \infty} U(t)^{-1}U_0(t)(\varphi)$$ exist.

I am having some trouble interpreting what the operators $\Omega_-$ and $\Omega_+$ represent. Talagrand provides some motivation in the pages before the proposition, and says that

$\Omega_+(x,v) = (x,v)_{\text{in}}$ represents the state at time $t = 0$ of the particle whose motion in the far past resembled the motion of a free particle of position and velocity $(x,v)$ at time $0$.

One defines similarly an operator $\Omega_-$ by starting with the free particle of position $x$ and velocity $v$ at time zero, running time forward until a large positive time $t_0$, turning the potential on, running time backward until we reach time zero again, and taking the limit as $t_0 \rightarrow \infty$. One also uses the notation $(x,v)_{\text{out}} = \Omega_-(x,v)$.

However I still am a little lost on what these $\Omega$ operators are doing exactly, and what is the physical reasoning behind defining them as well as the scattering operator $S = \Omega_-^{-1}\Omega_+$.