Structure of wave operators for a scaling-critical class of potentials

We prove a structure formula for the wave operators in ${\Bbb R}^3 $$W_{\pm} = \mathop{\hbox{s-lim}}\limits_{t\to\pm\infty} e^{it (-\Delta + V)} P_c e^{it\Delta}$$ and their adjoints for a scaling-invariant class of scalar potentials $V \in B$, $$B = \left\{V\mid\sum_{k\in\Bbb{Z}} 2^{k/2} \big\|\chi_{|x|\in [2^k, 2^{k+1}]}(x) V(x)\big\|_{L^2} < \infty\right\},$$ under the assumption that zero is neither an eigenvalue, nor a resonance for $-\Delta+V$. The formula implies the boundedness of wave operators on $L^p$ spaces, $1\leq p\leq\infty$, on weighted $L^p$ spaces, and on Sobolev spaces, as well as multilinear estimates for $e^{itH} P_c$. When $V$ decreases rapidly at infinity, we obtain an asymptotic expansion of the wave operators. The first term of the expansion is of order $\langle y \rangle^{-4}$, commutes with the Laplacian, and exists when $V \in \langle x \rangle^{-3/2-\epsilon} L^2$. We also prove that the scattering operator $S = W_-^* W_+$ is an integrable combination of isometries. The proof is based on an abstract version of Wiener's theorem, applied in a new function space.