On sec. 3.4.1 in Schlag & Nakanish: invariant manifolds and dispersive Hamiltionian evolution equations, the authors talked something about wave operators. The wave operators are defined as the strong limits in $L^2$, \begin{equation} W= s- \lim_{t \to \infty} e^{-it H } e^{it H_0}, \quad H=-\Delta+V, \; H=-\Delta. \end{equation} And I have two questions about it.
- If W exists, then W is isometric on $L^2$. By the fact that $W$ is isometric, how can I check that $WW^*$ and $W^*W$ are projections, the former onto $Ran(W)$, the latter being the identity?
- Can we precisely know $Ran (W)$ if we just know $W$ exists? Does $Ran(W)= L_c^2$, where $L_c^2$ is the continuous subspace w.r.t. $H=-\Delta+V$? Or it requires more on $V$, e.g., should we add more restriction onto $V$, such as add more decay and regularity onto $V$?