Wave Operators: Preliminary

Question

Given a Hilbert space $\mathcal{H}$.

Consider a free Hamiltonian $H_0$ and a perturbed one $H$.

Introduce the wave operators: $$\Omega^\pm(H,H_0):=\mathrm{s-lim}_{\tau\to\pm\infty}e^{i\tau H}e^{-i\tau H_0}P_\text{ac}(H_0)$$ (The existence being implicitely assumed!)

Why do we restrict to the absolutely continuous subspace?
(Would it work for the singular continuous subspace, too, or is it because of Riemann-Lebesgue?)

Answer 1

This following bits comes out of Reed and Simon Vol III, and I think this helps explain the problem. You should own Vol II and Vol III if you're seriously interested in this topic.

Definition: Suppose that $\Omega^{\pm}(A,B)$ exist. We say that they are complete if and only if $$ \mbox{Ran}\Omega^{+}=\mbox{Ran}\Omega^{-}=\mbox{Ran}P_{ac}(A). $$

This definition is followed immediately by

Proposition: Suppose that $\Omega^{\pm}(A,B)$ exist. Then they are complete if and only if $\Omega^{\pm}(B,A)$ exist.