Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians

Results concerning the rigorous justification of the effective Hamiltonians for band electrons in the presence of weak homogeneous electric and magnetic fields are reviewed. In the electric-field case the existence, in the sense of spectral concentration, of the Stark-Wannier resonances is proved. In the magnetic-field case, the existence of exponentially localized magnetic Wannier functions is established. As a consequence the Peierls-Onsager effective Hamiltonian is obtained.