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Floquet Energies and Quantum Hall Effect in a Periodic Potential

The Quantum Hall Effect for free electrons in external periodic field is discussed without using the linear response approximation. We find that the Hall conductivity is related in a simple way to Floquet energies (associated to the Schroedinger equation in the co-moving frame). By this relation one can analyze the dependence of the Hall conductivity from the electric field. Sub-bands can be introduced by the time average of the expectation value of the Hamiltonian on the Floquet states. Moreover we prove previous results in form of sum rules as, for instance: the topological character of the Hall conductivity (being an integer multiple of e^2/h), the Diofantine equation which constrains the Hall conductivity by the rational number which measures the flux of the magnetic field through the periodicity cell. The Schroedinger equation fixes in a natural way the phase of the wave function over the reduced Brillouin zone: thus the topological invariant providing the Hall conductivity can be evaluated numerically without ambiguity.