S-matrix

Remarks

The operator $S$ commutes with $H_0$, so it decomposes over the spectrum of $H_0$ into operators $S(E)$ on the energy shells, which are also called the scattering matrix. If $\operatorname{Ran} W_+ = \operatorname{Ran} W_-$, then $S$ is a unitary operator on the absolutely continuous subspace of $H_0$; this holds under asymptotic completeness. Example: if $H = H_0$, then $W_\pm = P_{ac}(H_0)$ and $S = P_{ac}(H_0)$, which is the identity there: no scattering. Example: let $H = -\Delta + V$ on $L^2(\R^3)$ with $V$ spherically symmetric and short range. Then $S(E)$ acts on the angular momentum $l$ component as multiplication by $e^{2i\delta_l(k)}$, with $k = \sqrt{E}$ and $\delta_l$ the phase shift. Non-example: if $\operatorname{Ran} W_- \ne \operatorname{Ran} W_+$, then $S$ is not unitary on the absolutely continuous subspace of $H_0$. Indeed, $S$ is an isometry there only if $\operatorname{Ran} W_- \subseteq \operatorname{Ran} W_+$, and a co-isometry only if the reverse inclusion holds. With the opposite sign convention for $W_\pm$, the same operator is written $S = W_-^* W_+$.