The Møller operator is defined as Ω+=limt→−∞U†(t)U0(t).
Does the operator
limt→−∞U†0(t)U(t)
also make sense? Is it important?
The overall idea is the following. You have a state evolving with the full interacting theory Ψ(t)=UtΨ. If the theory admits an asymptotic description for t→−∞, there must be a state Ψ0 evolving with the free theory Ψ0(t)=U0(t)Ψ0 such that the two evolutions ``coincide'' at large time in the past: limt→−∞||U0(t)Ψ0−U(t)Ψ||=0.
Using unitarity of
U you can write, form the above equation,
limt→−∞||U(t)†U0(t)Ψ0−Ψ||=0.
In other words
Ψ=Ω−Ψ0
(I prefer to define that operator as the Moller operator
Ω− instead of
Ω+) where
Ω−:=s -limt→−∞U(t)†U0(t)
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
Ω− is not the full Hilbert space
H, but just a closed subspace
H− because, in particular, there may exist bound states which do not admit a scattering description.
However (1) is also equivalent to limt→−∞||Ψ0−U0(t)†U(t)Ψ||=0.
and this define the inverse of
Ω− associating interacting states
Ψ∈H− to free states
Ψ0∈H. So
(Ω−)−1Ψ:=limt→−∞U0(t)†U(t)Ψ.
with
Ψ∈H−. (Here the inverse
(Ω−)−1 is the one of the isometric map
Ω−:H→H−.) That is
(Ω−)−1:=s−limt→−∞U0(t)†U(t).
where
s again denotes the
strong operator topology for a class of operators defined on the Hilbert space
H− and values in the Hilbert space
H. (
H− is closed due to a well-known elementary result on partial isometries in a Hilbert space. Since it is closed in
H,
H− is a Hilbert space in its own right.)
REMARKS
(1) The fact that the limit (4) exists for Ψ∈H−=Ran(Ω−) is evident from the fact that every sequence Ψ0−U0(tn)†U(tn)Ψ is Cauchy since the isometric sequence U(tn)†U0(tn)Ψ0−Ψ is Cauchy by hypotheses.
(2) The fact that the strong limit in (4) produces the inverse of Ω− on H− is equally obvious since Ω− associates Ψ0∈H to a certain Ψ∈H− and the limit operator s -limt→−∞U0(t)†U(t) associated that Ψ to the initial Ψ0.
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