  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.
