Scattering and perturbation theory for discrete-time dynamics

Understand

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.