Sobolev mapping properties of the scattering transform for the Schr\"odinger equation

We consider the scattering transform for the Schr\"odinger equation with a singular potential and no bound states. Using the Riccati representation for real-valued potentials on the line, we obtain invertibility and Lipschitz continuity of the scattering transform between weighted and Sobolev spaces. Our approach exploits the connection between scattering theory for the Schr\"odinger equation and scattering theory for the ZS-AKNS system