On Effective Hamiltonians for Adiabatic Perturbations of Magnetic   Schrodinger Operators
  We construct almost invariant subspaces and the corresponding effective Hamiltonian for magnetic Bloch bands. We also discuss the question of the dynamics related to the effective Hamiltonian. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and a periodic lattice potential, respectively. 
M. Dimassi, J.-C. Guillot, J. Ralston