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.