mathematical physics - About the definition of the Møller operator

The Møller operator is defined as $$\Omega_+ = \lim_{t\rightarrow -\infty} U^\dagger (t) U_0(t).$$ Does the operator $$ \lim_{t\rightarrow -\infty} U_0^\dagger (t) U(t) $$ also make sense? Is it important?

Answer

The overall idea is the following. You have a state evolving with the full interacting theory $\Psi(t) = U_t\Psi$. If the theory admits an asymptotic description for $t\to -\infty$, there must be a state $\Psi_0$ evolving with the free theory $\Psi_{0}(t) = U_0(t)\Psi_0$ such that the two evolutions ``coincide'' at large time in the past: $$\lim_{t\to -\infty}||U_0(t)\Psi_0- U(t) \Psi|| =0\:.$$ Using unitarity of $U$ you can write, form the above equation, $$\lim_{t\to -\infty}||U(t)^\dagger U_0(t) \Psi_0 - \Psi||=0\:.\tag{1}$$ In other words $$\Psi = \Omega_- \Psi_0$$ (I prefer to define that operator as the Moller operator $\Omega_-$ instead of $\Omega_+$) where $$\Omega_- := \mbox{s -}\lim_{t\to -\infty} U(t)^\dagger U_0(t) \tag{2}$$ that s denoting the strong operator topology involved in (1), assuming that there is an interacting scattering state for every free state of the Hilbert space. Usually the image of $\Omega_-$ is not the full Hilbert space $\cal H$, but just a closed subspace ${\cal H}_-$ because, in particular, there may exist bound states which do not admit a scattering description.

However (1) is also equivalent to $$ \lim_{t\to -\infty}||\Psi_0 - U_0(t)^\dagger U(t)\Psi||=0\:.\tag{3}$$ and this define the inverse of $\Omega_-$ associating interacting states $\Psi \in {\cal H}_-$ to free states $\Psi_0 \in \cal H$. So

$$(\Omega_-)^{-1} \Psi := \lim_{t\to -\infty} U_0(t)^\dagger U(t) \Psi\tag{4}\:.$$ with $\Psi \in {\cal H}_-$. (Here the inverse $(\Omega_-)^{-1}$ is the one of the isometric map $\Omega_- : {\cal H} \to {\cal H}_-$.) That is $$(\Omega_-)^{-1} := \mbox{s}-\lim_{t\to -\infty} U_0(t)^\dagger U(t) \tag{5}\:.$$ where $s$ again denotes the strong operator topology for a class of operators defined on the Hilbert space ${\cal H}_-$ and values in the Hilbert space ${\cal H}$. (${\cal H}_-$ is closed due to a well-known elementary result on partial isometries in a Hilbert space. Since it is closed in $\cal H$, ${\cal H}_-$ is a Hilbert space in its own right.)

REMARKS

(1) The fact that the limit (4) exists for $\Psi \in {\cal H}_-= Ran(\Omega_-)$ is evident from the fact that every sequence $\Psi_0 - U_0(t_n)^\dagger U(t_n) \Psi$ is Cauchy since the isometric sequence $U(t_n)^\dagger U_0(t_n) \Psi_0 - \Psi$ is Cauchy by hypotheses.

(2) The fact that the strong limit in (4) produces the inverse of $\Omega_-$ on ${\cal H}_-$ is equally obvious since $\Omega_-$ associates $\Psi_0 \in {\cal H}$ to a certain $\Psi \in {\cal H}_-$ and the limit operator $\mbox{s -}\lim_{t\to -\infty} U_0(t)^\dagger U(t)$ associated that $\Psi$ to the initial $\Psi_0$.