  Some relevant transport properties of solids do not depend only on the
spectrum of the electronic Hamiltonian, but on finer properties preserved only
by unitary equivalence, the most striking example being the conductance. When
interested in such properties, and aiming to a simpler model, it is mandatory
to check that the simpler effective Hamiltonian is approximately unitarily
equivalent to the original one, in the appropriate asymptotic regime. In this
paper, we consider the Hamiltonian of an electron in a 2-dimensional periodic
potential (e.g. generated by the ionic cores of a crystalline solid) under the
influence of a uniform transverse magnetic field. We prove that such
Hamiltonian is approximately unitarily equivalent to a Hofstadter-like (resp.
Harper-like) Hamiltonian, in the limit of weak (resp. strong) magnetic field.
The result concerning the case of weak magnetic field holds true in any
dimension. Finally, in the limit of strong uniform magnetic field, we show that
an additional periodic magnetic potential induces a non-trivial coupling of the
Landau bands.
