In the majority of textbooks on quantum physics, the apparently
straightforward presentation of quantum scattering proceeds via wave
operators $W_\pm$ defined in terms of certain limits of resolvents.
It is well known (cf., e.g., [25]) that the underlying rigorous
mathematical analysis is fairly subtle. In this context, the MS in
question underlines the relevance of some of these subtleties,
taking, for the particularly difficult correct description of the
centrally asymmetric scattering in two-dimensional space, advantage
of important assumptions of a quick growth of the interaction
potential ($|V(x)|\leq {\rm Const.}\;\!(1+|x|)^{-\sigma}$ with
$\sigma>11$ for almost every $x\in{\mathbb R}^2$) and of a
``generic'' absence of eigenvalues or of resonances at zero energy.
The main result (theorem 1.1) are formulae $W_-=1+R(A)(S-1)+K$ in
${\mathscr B}({\mathcal H})$ (bounded operators - plus a similar one
for $W_+$) where symbol $A$ stands for the generator of dilations in
${\mathbb R}^2$ while $R(A):=\frac{1}{2}\left(1+\tanh(\pi
A/2)\right)$. The compactness of $K$ (from an ideal) is emphasized
to establish connections to the reinterpretation of the Levinson's
theorem as an index theorem.


MR3089680 Richard, Serge; Tiedra de Aldecoa, Rafael Explicit
formulas for the Schrödinger wave operators in $\Bbb R\sp 2$. C. R.
Math. Acad. Sci. Paris 351 (2013), no. 5-6, 209--214. 47F05 (35J10
35P25)