  We present a systematic treatment of scattering processes for quantum systems
whose time evolution is discrete. We define and show some general properties of
the scattering operator, in particular the conservation of quasi-energy which
is defined only modulo $2 \pi$. Then we develop two perturbative techniques for
the power series expansion of the scattering operator, the first one analogous
to the iterative solution of the Lippmann-Schwinger equation, the second one to
the Dyson series of perturbative Quantum Field Theory. We use this formalism to
compare the scattering amplitudes of a continuous-time model and of the
corresponding discretized one. We give a rigorous assessment of the comparison
for the case of bounded free Hamiltonian, as in a lattice theory with a bounded
number of particles. Our framework can be applied to a wide class of quantum
simulators, like quantum walks and quantum cellular automata. As a case study,
we analyse the scattering properties of a one-dimensional cellular automaton
with locally interacting fermions.
