The Møller operator is defined as Ω+=lim 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.
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