Wave Operators: Unitarity This thread is Q&A.
Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$.
Consider Hamiltonians:
$$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$
Denote their evolutions:
$$U_\#(t)^*=U_\#(-t)=U_\#(t)^{-1}$$
For a bounded operator:
$$J:\mathcal{H}_0\to\mathcal{H}:\quad\|J\|<\infty$$
Assume the limit:
$$\Omega\varphi:=\lim_{t\to\infty}U(t)^*JU_0(t)\varphi\quad(\varphi\in\mathcal{H})$$
Polar decomposition:
$$\Omega=J_\Omega|\Omega|:\quad\mathcal{N}J_\Omega=\left(\overline{\mathcal{R}|\Omega|}\right)^\perp$$

Then one has:
  $$\eta\in\mathcal{B}(\mathbb{R}):\quad J_\Omega\eta(H_0)\subseteq\eta(H)J_\Omega$$

How can I prove this?
 A: Denote for shorthand:
$$\Omega':=\Omega^*\Omega\in\mathcal{B}(\mathcal{H}):\quad |\Omega|=\lim_n\Omega'_n$$
Regard spectral measures:
$$H=\int\lambda\mathrm{d}E(\lambda)\quad H_0=\int\lambda\mathrm{d}E_0(\lambda)$$
By a previous thread:*
$$\Omega E_0(A)=E(A)\Omega\implies\Omega'E_0(A)=E_0(A)\Omega'\implies(\Omega')^n E_0=E_0(A)(\Omega')^n$$
So one arrives at:
$$E_0(A)|\Omega|=E_0(A)\lim_n\Omega'_n=\lim_nE_0(A)\Omega'_n=\lim_n\Omega'_nE_0(A)=|\Omega|E_0(A)$$
So one obtains:
$$J_\Omega E_0(A)|\Omega|=J_\Omega|\Omega|E_0(A)=\Omega E_0(A)=E_0(A)\Omega=E(A)J_\Omega|\Omega|$$
Regard an element:
$$\psi=\lim_n|\Omega|\varphi_n\in\overline{\mathcal{R}|\Omega|}$$
By continuity one has:
$$J_\Omega E_0(A)\psi=J_\Omega E_0(A)\lim_n|\Omega|\varphi_n=\lim_nJ_\Omega E_0(A)|\Omega|\varphi_n\\
=\lim_nE_0(A)J_\Omega|\Omega|\varphi_n=E(A)J_\Omega\lim_n|\Omega|\varphi_n=E(A)J_\Omega\psi$$
Regard an element:
$$\psi\in\left(\overline{\mathcal{R}|\Omega}|\right)^\perp=\mathcal{N}|\Omega|=\mathcal{N}\Omega=\mathcal{N}J_\Omega$$
By reducibility one has:**
$$J_\Omega E_0(A)\psi=J_\Omega1_{(\mathcal{N}\Omega)^\perp}E_0(A)\psi=J_\Omega E_0(A)1_{(\mathcal{N}\Omega)^\perp}\psi=0=E(A)J_\Omega\psi$$
By measurable calculus:***
$$J_\Omega E_0(A)=E(A)J_\Omega\quad(A\in\mathcal{B}(\mathbb{R}))\implies J_\Omega\eta(H_0)\subseteq\eta(H)J_\Omega\quad(\eta\in\mathcal{B}(\mathbb{R}))$$
Concluding the assertion.
*See the thread: Calculus (WO)
**See the thread: Reducibility (WO)
***See proof of: Reducibility (SM)
