  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.
