  We study the scattering of an electron from a one dimensional inverted
Gaussian atomic potential in the presence of strong time periodic electric
fields. Using Floquet theory, we construct the Floquet Scattering matrix in the
Kramers-Henneberger frame. We compute the transmission coefficients as a
function of electron incident energy and find that they display asymmetric Fano
resonances due to the electron interaction with the driving field. We find that
the Fano resonances are associated with zero-pole pairs of the Floquet
Scattering matrix in the complex energy plane. Another way we "probe" the
complex spectrum of the system is by computing the Wigner-Smith delay times.
Finally we find that the eigenphases of the Floquet Scattering matrix undergo a
number of "avoided crossings" as a function of electron Floquet energy that
increases with increasing strength of the driving field. These "avoided
crossings" appear to be quantum manifestations of the destruction of the
constants of motion and the onset of chaos in classical phase space.
