Scattering theory

Remarks

When they exist, $W_\pm$ are partial isometries from the absolutely continuous subspace of $H_0$ onto their ranges, and they satisfy $e^{isH} W_\pm = W_\pm e^{isH_0}$ for all $s \in \R$. Some authors use the opposite sign convention, with $W_\pm$ defined by the limit $t \to \mp\infty$. Example: $\mathcal{H} = L^2(\R^3)$, $H_0 = -\Delta$ and $H = -\Delta + V$ with $V$ real and $\abs{V(x)} \le C(1 + \abs{x})^{-1-\varepsilon}$ for some $C, \varepsilon > 0$; then the wave operators exist and are asymptotically complete. Non-example: for the Coulomb potential $V(x) = c/\abs{x}$ with $c \ne 0$, the limits in clause 1 do not exist. One needs modified wave operators, introduced by Dollard, which correct the free dynamics by a logarithmic phase. The unitary groups $e^{-itH}$ and $e^{-itH_0}$ are the solutions of the time-dependent Schrödinger equations of $H$ and $H_0$.