Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2

For the Schrödinger equation in dimension 2 we reconstruct the potential $v\in W^{N,1}_{\varepsilon}(\mathbb R^2)$, $\mathbb N\ni N\ge 3$, $\varepsilon>0$ ($N$-times smooth potential) from the scattering amplitude $f$ at fixed energy $E$ up to $O(E^{-(N-2)/2})$ in the uniform norm as $E\to+\infty$.