Wave Operators: Cook

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$.

Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$

Denote their evolutions: $$U_\#(t)^*=U_\#(-t)=U_\#(t)^{-1}$$

Regard an operator: $$J:\mathcal{H}_0\to\mathcal{H}:\quad\|J\|<\infty$$

Suppose invariance: $$J\mathcal{D}H_0^2\subseteq\mathcal{D}H^2$$

And the estimate: $$\int_0^\infty\|\{HJ-JH_0\}U_0(s)\varphi\|\mathrm{d}s<\infty\quad(\varphi\in\mathcal{D}H_0^2)$$

Then the limit exists: $$\Omega\varphi:=\lim_{t\to\infty}U(t)^*JU_0(t)\varphi\quad(\varphi\in\mathcal{H})$$

How can I prove this?

Meanwhile I got it...

Define the function: $$\varphi\in\mathcal{D}H_0^2:\quad\omega(t):=U(t)^*JU_0(t)\varphi$$

Its derivatives are: $$\omega'(t)=U(t)i\{HJ-JH_0\}U_0(t)\varphi$$ $$\omega''(t)=U(t)i^2\{H^2J-2HJH_0+JH_0^2\}U_0(t)\varphi$$

By Pettis' criterion:* $$\omega''\in\mathcal{F}(\mathbb{R},\mathcal{H})\implies\omega'\in\mathcal{C}(\mathbb{R},\mathcal{H})\implies\omega'\in\mathcal{B}(\mathbb{R},\mathcal{H})$$

And it is integrable: $$\int_0^\infty\|\omega'(s)\|\mathrm{d}s=\int_0^\infty\|\{HJ-JH_0\}U_0(s)\varphi\|\mathrm{d}s<\infty$$

By fundamental theorem: $$U(t)^*JU_0(t)\varphi-J\varphi=\omega(t)-\omega(0)=\int_0^t\omega'(s)\mathrm{d}s$$

By dominated convergence: $$\|U(t')^*JU_0(t')\varphi-U(t)^*JU_0(t)\varphi\|=\|\int_t^{t'}\omega'(s)\mathrm{d}s\|\stackrel{t,t'\to\infty}{\to}0$$

But the domain is dense: $$\overline{\mathcal{D}H_0^2}\supseteq\overline{\bigcup_{R>0}\mathcal{R}E(B_R)}=\mathcal{H}_0$$

Concluding the assertion.