Equivalence of Topological and Scattering Approaches to Quantum Pumping

The Schrodinger equation with a potential periodically varying in time is used to model adiabatic quantum pumps. The systems considered may be either infinitely extended and gapped or finite and connected to gapless leads. Correspondingly, two descriptions of the transported charge, one relating to a Chern number and the other to a scattering matrix, have been available for some time. Here we generalize the first one and establish its equivalence to the second.