Inverse scattering at a fixed quasi-energy for potentials periodic in time

We prove that the scattering matrix at a fixed quasi-energy determines uniquely a time-periodic potential that decays exponentially at infinity. We consider potentials that for each fixed time belong to L-3/2 in space. The exponent 3/2 is critical for the singularities of the potential in space. For this singular class of potentials the result is new even in the time-independent case, where it was only known for bounded exponentially decreasing potentials.