Wave Operators: Summary

Question

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

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

Then one obtains: $$\big(H\restriction_\overline{\mathcal{R}\Omega}\big)=\big(H\restriction_\overline{\mathcal{R}\Omega}\big)^*\cong\big(H_0\restriction_\overline{\mathcal{R}\Omega^*}\big)=\big(H_0\restriction_\overline{\mathcal{R}\Omega^*}\big)^*$$

How can I prove this?

Answer 1

Note that one has: $$(\mathcal{N}\Omega)^\perp=\overline{\mathcal{R}\Omega^*}=\overline{\mathcal{R}|\Omega|}$$

Denote embeddings: $$J^0_\overline{\mathcal{R}\Omega^*}\in\mathcal{B}\big(\overline{\mathcal{R}\Omega^*},\mathcal{H}_0\big):\quad\big(J^0_\overline{\mathcal{R}\Omega^*}\big)^*\big(J^0_\overline{\mathcal{R}\Omega^*}\big)=1_\overline{\mathcal{R}\Omega^*}$$ $$J_\overline{\mathcal{R}\Omega}\in\mathcal{B}\big(\overline{\mathcal{R}\Omega},\mathcal{H}\big):\quad\big(J_\overline{\mathcal{R}\Omega}\big)^*\big(J_\overline{\mathcal{R}\Omega}\big)=1_\overline{\mathcal{R}\Omega}$$

Restrict Hamiltonians: $$H^0_\overline{\mathcal{R}\Omega^*}:=\big(J^0_\overline{\mathcal{R}\Omega^*}\big)^*H_0\big(J^0_\overline{\mathcal{R}\Omega^*}\big)\quad H_\overline{\mathcal{R}\Omega}=\big(J_\overline{\mathcal{R}\Omega}\big)^*H\big(J_\overline{\mathcal{R}\Omega}\big)$$

By reducibility one has:* $$\big(H^0_\overline{\mathcal{R}\Omega^*}\big)=\big(H^0_\overline{\mathcal{R}\Omega^*}\big)^*\quad\big(H_\overline{\mathcal{R}\Omega}\big)=\big(H_\overline{\mathcal{R}\Omega}\big)^*$$

Polar decomposition: $$\Omega=J_\Omega|\Omega|:\quad(J_\Omega)^*(J_\Omega)=1_\overline{\mathcal{R}\Omega^*}\quad(J_\Omega)(J_\Omega)^*=1_\overline{\mathcal{R}\Omega}$$

Denote unitary map: $$U_\Omega:\overline{\mathcal{R}\Omega^*}\to\overline{\mathcal{R}\Omega}:\quad U_\Omega\varphi:=J_\Omega\varphi$$

By unitarity one has:** $$\big(J_\overline{\mathcal{R}\Omega}\big)^*U(t)\big(J_\overline{\mathcal{R}\Omega}\big)\big(U_\Omega\big)=\big(U_\Omega\big)\big(J^0_\overline{\mathcal{R}\Omega^*}\big)^*U_0(t)\big(J^0_\overline{\mathcal{R}\Omega^*}\big)$$

By Stone's theorem: $$\big(H_\overline{\mathcal{R}\Omega}\big)=(U_\Omega)\big(H^0_\overline{\mathcal{R}\Omega^*}\big)(U_\Omega)^*$$

Concluding equivalence.

*See the thread: Reducibility

**See the thread: Unitarity